# Robust Fault Estimation for a Class of T-S Fuzzy Singular Systems with Time-Varying Delay via Improved Delay Partitioning Approach

Authors: Chao Sun; FuLi Wang; XiQin He
Journal: Journal of Control Science and Engineering
 (2016)

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## Abstract

The problem of delay-dependent robust fault estimation for a class of Takagi-Sugeno (T-S) fuzzy singular systems is investigated. By decomposing the delay interval into two unequal subintervals and with a new and tighter integral inequality transformation, an improved delay-dependent stability criterion is given in terms of linear matrix inequalities (LMIs) to guarantee that the fuzzy singular system with time-varying delay is regular, impulse-free, and stable firstly. Then, based on this criterion, by considering the system fault as an auxiliary disturbance vector and constructing an appropriate fuzzy augmented system, a fault estimation observer is designed to ensure that the error dynamic system is regular, impulse-free, and robustly stable with a prescribedH∞ performance satisfied for all actuator and sensor faults simultaneously, and the obtained fault estimates can practically better depict the size and shape of the faults. Finally, numerical examples are given to show the effectiveness of the proposed approach.


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## Body

## 1. Introduction

The demand for increased productivity leads to more challenging operating conditions for many modern engineering systems. The issue of fault detection and isolation (FDI) algorithms in dynamic systems and their applications to a wide range of industrial processes have been an active research area over the past two decades, as can be seen, in survey papers [1] for linear systems, [2] for multimodels representation and [3, 4] for nonlinear systems. By using FDI procedures, the reliability can be achieved by fault-tolerant control, which relies on early detection and isolation of faults. So FDI have become a popular topic and received considerable attention. However, it is generally difficult, in practice engineering, to obtain the exact information of the size of system fault from an FDI strategy only because of the existence of model uncertainties, time delays, and disturbances [5]. As pointed out in [6], accurate and timely fault estimation is an important antecedent for satisfactory control reconfiguration. Therefore, the problem of fault estimation has stirred renewed research interest, and a variety of fault estimation approaches have been developed in the literatures; see, for example, [7–11] and the references therein.On the other hand, singular systems have been extensively studied in the past years due to the fact that singular systems better describe physical systems than state-space ones, especially the T-S fuzzy singular systems, because they can combine the flexibility of fuzzy logic theory and fruitful linear singular system theory into a unified framework to approximate complex nonlinear singular systems. In fact, singular systems can be found in electrical circuits, economic systems, moving robots, and many other systems. Recently, many results about fault estimation have been reported on analysis and design of singular systems. For instance, by using online learning methodology, [12] proposed a fault estimation method for continuous-time nonlinear singular systems. Reference [7] used generalized unknown input observer to deal with the robust fault detection problem for linear singular systems. Reference [13] deals with actuator fault estimation for a class of discrete-time linear parameter-varying singular systems. Reference [14] designed a robust fault detection filter for a class of nonlinear singular systems described by linear parameter-varying form with global Lipschitz term. However, it is known that time delays are frequently encountered in various engineering and communication systems, and a time delay in a dynamical system is often a primary source of instability and performance degradation. Therefore, it is important to develop fault estimation methods for time delay singular systems. But the fault diagnosis for singular systems with time-varying delay has not been well investigated yet [7, 12–14]. More recently, [15] proposed a fault detection, isolation, and estimation scheme via unknown input proportional integral observers for linear descriptor systems. Reference [8] investigated fault detection for discrete-time switched singular systems with time-varying delays. It should be noticed that [8] deals with discrete-time switched singular systems while this paper focuses on the fuzzy continuous-time case. In [16], discrete-time T-S systems with sensor faults are first formulated as a descriptor representation, and then a fault detection filter is designed based on the obtained descriptor system. However, this paper studies fault detection for regular systems by using the technique of descriptor systems. In [17], the author proposed a k-step fault estimation method for T-S fuzzy time delay system that only deals with regular systems. Moreover, our paper considers how to estimate the actuator and sensor faults simultaneously while attenuating the influence of the disturbance noise, which is not considered in [17]. It is known that singular system representation is a generalization of the regular system. Therefore, the proposed method is more general than that in [16, 17].The aim of this paper is to develop a robust fault estimation method for a class of T-S fuzzy singular systems with time delays. The basic idea is to construct an augmented system by taking the fault as auxiliary disturbance vector and then design a fault estimation observer based on this augmented system. The main contribution of the proposed method lies in the following aspects. First, without ignoring any useful terms in the Lyapunov-Krasovskii functional (LKF), by decomposing the delay interval into two unequal subintervals, a new LKF is constructed; then the free weighting matrices approach is introduced to develop a new delay-dependent stability criteria, which ensure that the considered system is regular, impulse-free, and stable. Compared with some existing results, the approach to be proposed in this paper can be expected to give better results. Second, a new robust fault estimation observer with a novel structure is proposed for T-S fuzzy singular systems with time delays and actuator and sensor faults simultaneously, which is the main contribution of this paper. The proposed observer can be designed by solving a set of linear matrix inequalities and to attenuate the effect of unknown disturbance, fault variation on fault estimation. The effectiveness of the method is illustrated by some numerical examples.The rest of this paper is organized as follows. The system description and preliminaries are presented in Section2. Section 3 presents the main results on new stability criteria of fuzzy singular systems with time-varying delays and robust fault estimation observer design scheme. In Section 4, simulation results of numerical examples are presented to demonstrate the effectiveness and merits of the proposed methods. Finally, Section 5 concludes the paper.Notations. Throughout the paper, Rn denotes the n-dimensional real Euclidean space; I denotes the identity matrix; the superscripts T and -1 stand for the matrix transpose and inverse, respectively; notation X>0(X≥0) means that matrix X is real symmetric positive definite (positive semidefinite); · is the spectral norm. If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations. The symbol “∗” stands for matrix block induced by symmetry; sym(X) stands for X+XT.


## 2. System Description and Preliminaries

Consider a nonlinear singular system which can be represented by the following extended T-S fuzzy time delay model with external disturbance, actuator and sensor faults, simultaneously.Plant Rule
i. If ξ1(t) is Mi1 and … and ξp(t) is Mip, then(1)Ex˙t=Aixt+Aτixt-τt+Biut+Bfift+Bdidt,yt=Cixt+Cτixt-τt+Diut+Dfift+Ddidt,xt=ϕit,∀t∈-τ2,0,i=1,2,…,r,where x(t)∈Rn is the state vector, u(t)∈Rq denotes the input vector, and y(t)∈Rl stands for the system output vector. d(t)∈Rm is the exogenous disturbance input that belongs to L2[0,∞); f(t)∈Rq represents the possible fault. The matrix E∈Rn×n is a constant matrix, which may be singular; that is, rank(E)=g≤n. Ai, Aτi, Bi, Bfi, Bdi, Ci, Cτi, Di, Dfi, and Ddi are constant real matrices of appropriate dimensions. It is assumed that the pairs (E,Ai,Ci) are of full column rank, where i=1,2,…,r. ξ1(t),…,ξp(t) are the premise variables, Mij(i=1,2,…,r,j=1,2,…,p) are fuzzy sets, and ϕi(t) is a vector-valued initial continuous function defined on the interval [-τ2,0]. In this paper, it is also assumed that the premise variables do not depend on the input variables u(t); τ(t) is the time-varying delay and satisfies(2)τ1≤τt≤τ2,0≤τ˙t≤τD.Then, by fuzzy blending, the overall fuzzy singular system model is given by(3)Ex˙t=∑i=1rμiξtAixt+Aτixt-τt+Biut+Bfift+Bdidt,yt=∑i=1rμiξtCixt+Cτixt-τt+Diut+Dfift+Ddidt,xt=∑i=1rμiξtϕit,∀t∈-τ2,0,where the fuzzy basis functions are given by (4)μiξt=βiξt∑j=1rβjξt,βiξt=∏i=1pMijξt where Mij(ξj(t)) represents the grade of membership of ξj(t) in Mij. Here, it is easy to find that for all t>0, we have(5)βiξt≥0,i=1,2,…,r,∑j=1rβjξt>0μiξt≥0,i=1,2,…,r,∑j=1rμjξt=1 For convenience of notations, in the sequel, we denote (6)At=∑i=1rμiξtAi,Aτt=∑i=1rμiξtAτi,Bt=∑i=1rμiξtBi,Bft=∑i=1rμiξtBfi,Bdt=∑i=1rμiξtBdi,Ct=∑i=1rμiξtCi,Cτt=∑i=1rμiξtCτi,Dt=∑i=1rμiξtDi,Dft=∑i=1rμiξtDfi,Ddt=∑i=1rμiξtDdi,ϕt=∑i=1rμiξtϕit. Then, we can rewrite system (3) as(7)Ex˙t=Atxt+Aτtxt-τt+Btut+Bftft+Bdtdt,yt=Ctxt+Cτtxt-τt+Dtut+Dftft+Ddtdt,xt=ϕt,t∈-τ2,0.Before proceeding further, we will introduce some definitions and assumptions to be needed in the development of main results throughout this paper. Consider an unforced singular time delay system described by(8)Ex˙t=Axt+Aτxt-τtxt=ϕt,t∈-τ,0.Definition 1 (see [18]).
(1) The pair(E,A) is said to be regular if det(zE-A) is not identically zero.
(2) The pair(E,A) is said to be impulse-free if deg(det(zE-A))=rank(E).
(3) The pair(E,A) is said to be stable, if all roots of det(zE-A)=0 lie inside the unit disk with center at the origin.
(4) The delayed singular system (8) is said to be admissible if the pair (E,A) is regular, impulse-free, and stable.Definition 2 (see [19]).
(1) The singular system (8) is said to be regular and impulse-free if the pair (E,A) is regular and impulse-free.
(2) The singular system (8) is said to be asymptotically stable, if for any ɛ>0, there exists a scalar δ(ɛ)>0 such that, for any compatible initial conditions, ϕ(t) with sup-τ(t)≤t≤0ϕ(t)<δ(ɛ); the solution x(t) of (8) satisfies x(t)<ɛ for t≥0 and limt→∞x(t)=0.
The singular time delay system (8) may have an impulsive solution. However, the regularity and nonimpulse of (E,A) guarantee the existence and uniqueness of impulse-free solution to (8) on [0,∞).Lemma 3 (see [20]).
If a functionalV:Cn[-τ,0]→R is continuous and x(t,ϕ) is a solution to (8), one defines V˙(ϕ)=limh→0+sup1/h(V(x(t+h,ϕ)-V(ϕ))). Denote the system parameters of (8) as (9)E,A,Aτ=Ig000,A11A12A21A22,Aτ11Aτ12Aτ21Aτ22.Assume that the singular system (8) is regular and impulse-free, A22 is invertible, and ρ(A22-1Aτ22)<1. Then, system (8) is stable if there exists positive numbers α, μ, ν, and a continuous function; V:Cn[-τ,0]→R, such that (10)μϕ102≤Vϕ≤νϕ2,V˙xt≤-αxt2,where xt=x(t+θ) with θ∈[-τ,0] and ϕ=[ϕ1Tϕ2T] with ϕ1∈Rq.In order to address the main results, the following assumptions are made.Assumption 4.
MatricesE and Ci satisfy the following rank condition: (11)rank⁡ECi=n.Assumption 5.
The triple matrix(E,Ai,Ci) is R-detectable [21] for ∀i=1,2,…,r; that is, (12)rank⁡sE-AiCi=n,s∈C+.Remark 6.
Both Assumptions4 and 5 are necessary conditions for the existence of the designed observer in the latter section. Similar assumptions can be also found in [21] and the references therein. Meanwhile, for T-S fuzzy system description (3), we can see that a general time-varying delay fuzzy singular system is considered in this paper, including possible actuator, sensor faults, and exogenous disturbance input simultaneously.


## 3. Main Results

### 3.1. Delay-Dependent Stability

In this subsection, we suggest developing a delay-dependent stability condition for the nominal unforced fuzzy singular system of (7), which can be written as(13)Ex˙t=Atxt+Aτtxt-τt+Bdtdt,yt=Ctxt+Cτtxt-τt+Ddtdt,xt=ϕt,t∈-τ2,0.Theorem 7.
For the givenτ1, τ2, τD, the free fuzzy singular system (13) with d(t)=0 is admissible for any time-varying delay τ(t) satisfying (2), if there exists a nonsingular matrix P, symmetric positive-definite matrices Q1>0, Q2>0, S1>0, S2>0, and R>0, and any appropriately dimensioned matrices V=V1V2V3V4V5 and W=W1W2W3W4W5, such that the following set of inequalities hold:(14)ETP=PTE≥0,(15)Ψiτ2-τ1VT∗-τ2-τ1R<0,(16)Ψiτ2-τ1WT∗-τ2-τ1R<0,where (17)Ψi=Ψ1i+τ2-τ1Γ1TRΓ1,Ψ1i=Ψ11i-V1TEV1TE-W1TEW1TEPTAτi∗Ψ22Ψ23W2TE-ETV4-ETV5∗∗Ψ33Ψ34ETV5-ETW5∗∗∗Ψ44ETW5∗∗∗∗-ɛ1-τDQ1 with (18)Ψ11i=PTAi+AiTP+ɛQ1+Q2,Ψ22=S1-Q2-V2TE-ETV2,Ψ23=V2TE-W2TE-ETV3,Ψ33=S2-S1+sym⁡V3TE-W3TE,Ψ34=W3TE-ETV4-ETW4,Ψ44=-S2+W4TE+ETW4.Proof.
The proof of this theorem is divided into two parts. The first one is concerned with the regularity and the impulse-free characterizations, and the second one treats the stability property of system (13).
Since rank(E)=g≤n, there must exist two invertible matrices G∈Rn×n and H∈Rn×n such that(19)E~=GEH=Ig000.Similar to (19), we define(20)A~i=GAiH=GAi11Ai12Ai21Ai22H=A~i11A~i12A~i21A~i22,i=1,2,…,r,P~=G-TPH=G-TP11P12P21P22H=P~11P~12P~21P~22.Since Ψi<0 and Q1>0 and Q2>0 and R>0, we can formulate the following inequality easily:(21)Ϝ=AiTP+PTAi<0.Then, pre- and postmultiplying Ϝ<0 by HT and H, respectively, (21) yields(22)Ϝ~=A~iTP~+P~TA~i=Ϝ~11Ϝ~12∗A~i22TP~22+P~22TA~i22<0.Since Ϝ~11 and Ϝ~12 are irrelevant to the results of the following discussion, the real expression of these two variables is omitted here. From (22), it is easy to see that (23)A~i22TP~22+P~22TA~i22<0. Since μi(ξ(t))≥0 and ∑i=1rμi(ξ(t))=1, we have (24)∑i=1rμiξtA~i22TP~22+P~22TA~i22<0. This implies that ∑i=1rμi(ξ(t))A~i22 is nonsingular. Therefore, the unforced fuzzy singular system (13) is regular and impulse-free.
Next, we will show the stability of system (13). If conditions (15)-(16) hold, we have (25)∑i=1rμiPTAi+AiTP+ɛQ1+Q2PTAτi∗-ɛ1-τDQ1<0. Premultiplying and postmultiplying the preceding inequality by (26)0I00000I and its transposes, respectively, we obtain (27)sym⁡P22T∑i=1rμiAi22+ɛQ122+Q222P22T∑i=1rμiAτi22∗-ɛ1-τDQ122<0. which implies that(28)
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Now, let us choose the following Lyapunov-Krasovskii function as(31)Vt=V1t+V2t+V3t, where(32)V1t=xTtETPxt,V2t=ɛ∫t-τttxTsQ1xsds+∫t-τ1txTsQ2xsds+∫t-τρt-τ1xTsS1xsds+∫t-τ2t-τρxTsS2xsds,V3t=∫-τ2-τ1∫t+θtx˙TsETREx˙sdsdθ, where the unknown matrices P, Q1>0, Q2>0, S1>0, S2>0, and R>0 are to be determined, and τρ=τ1+ρδ, δ=τ2-τ1, and 0<ρ<1. Then, the time derivatives of V(t) along the trajectories of the dynamic systems (13) satisfy (33)V˙1t=xTtPTAt+ATtPxt+2xTtPTAτtxt-τt,V˙2t=ɛxTtQ1xt-ɛ1-τ˙txTt-τtQ1xt-τt+xTtQ2xt-xTt-τ1Q2xt-τ1+xTt-τ1S1xt-τ1-xTt-τρS1xt-τρ+xTt-τρS2xt-τρ-xTt-τ2S2xt-τ2,V˙3t=τ2-τ1Ex˙tTREx˙t-∫t-τ2t-τ1x˙TsETREx˙sds=τ2-τ1Ex˙tTREx˙t-∫t-τρt-τ1x˙TsETREx˙sds-∫t-τ2t-τρx˙TsETREx˙sds. Denoting Γ1=A(t)000Aτ(t),  ζ1T(t)=xT(t)xT(t-τ1)xT(t-τρ)xT(t-τ2)xT(t-τ(t)) and β1Tt,s=ζ1T(t)(Ex˙(s))T, and from Newton-Leibniz formula, we can easily obtain that(34)-2ζ1TtVTExt-τ1-Ext-τρ-∫t-τρt-τ1Ex˙sds=0,-2ζ1TtWTExt-τρ-Ext-τ2-∫t-τ2t-τρEx˙sds=0. Therefore, a straightforward computation gives (35)
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In some existing literature, for example, [22, 23], some delay-dependent criteria are given in terms of LMIs to guarantee that the fuzzy singular system is admissible by using LKF approach and integral inequality, such as Lemma  2 in [22] and Lemma  2.3 in [23]. However, in the proof of our result of Theorem 7, we use one identical equality to estimate the upper bound of the derivative of V(t) without any model transformation. Moreover, it is interesting to mention that this study presents criteria based on the free weighting matrix method, in which the bounding techniques on some cross product terms are not involved [11, 23]. The major feature of this method is to reduce the conservatism engendered by the system transformations and the bounding techniques.Remark 9.
For time delay systems, the Lyapunov functional candidate always involves the integral term∫-τ0∫t+θtx˙TsRx˙sdsdθ, and the derivative of it was estimated as τx˙T(t)Rx˙(t)-∫t-τtx˙T(s)Rx˙(s)ds. However, the term ∫t-τtx˙T(s)Rx˙(s)ds was ignored [25–27], or some useful negative integral term was lost; see, for example, [28, 29]. Instead, in this paper all those terms ∫t-τρt-τ1x˙T(s)Rx˙(s)dsand ∫t-τ2t-τρx˙T(s)Rx˙(s)ds, which contain a great amount of useful information about systems, are preserved. Therefore, it is obvious to see that this method will lead to less conservatives than the existing ones in [25–29]. Furthermore, the introduction of parameter ɛ(ɛ≥0) indicates that Lemma 3 can be suitable for time-varying delay τ(t) being unknown or not differentiable; that is, in the case of time-varying delay τ(t) not differentiable, one can set ɛ=0.In order to estimate system faults, the following fault estimation observer is constructed:(36)Ex^˙t=Atx^t+Aτtx^t-τt+Btut+Bftf^t-Lty^t-yt,y^t=Ctx^t+Cτtx^t-τt+Dtut+Dftf^t,f^˙t=-Fty^t-yt,where x^(t)∈Rn is the observer state, y^(t)∈Rl is the observer output, and f^(t)∈Rq is an estimate of fault f(t). The objective is to design the appropriate dimension gain matrices L(t)∈Rn×l and F(t)∈Rq×l and estimate the fault despite the presence of the disturbance and state delay, where L(t)=∑i=1rμi(ξ(t))Li and F(t)=∑i=1rμi(ξ(t))Fi. Let us define ex(t)=x^(t)-x(t), ey(t)=y^(t)-y(t), ef(t)=f^(t)-f(t), and eT(t)=[exT(t),efT(t)], ωT(t)=[dT(t),f˙T(t)]; then the error dynamic systems are deduced from (7) and (36) as follows:(37)E¯e˙t=A¯t-L¯tC¯tet+A¯τt-L¯tC¯τtet-τt+L¯tD¯dt-B¯dtωt,eyt=C¯tet+C¯τtet-τt-D¯dtωt,where (38)E¯=E00Iq,A¯t=AtBft00,A¯τt=Aτt000,B¯dt=Bdt00Iq,L¯t=LtFt,C¯t=CtDft,C¯τt=Cτt0,D¯dt=Ddt0. Therefore, H∞ robust fault estimation observer design problem to be addressed in this paper can be formulated as follows: (i) The error dynamic system (37) with ω(t)=0 is admissible for any time delay satisfying (2); (ii) for a given scalar γ, the following H∞ performance is satisfied:(39)∫0Left2dt≤γ2∫0Lωt2dtfor all L>0 and ω(t)∈L2[0,∞) under zero initial conditions.For simplicity, we introduce the following vectors:(40)ζ2Tt=eTteTt-τ1eTt-τρeTt-τ2eTt-τtωTt,Γ2=A¯t-L¯tC¯t000A¯τt-L¯tC¯τtL¯tD¯dt-B¯dt. Then, the state of error dynamics (37) can be rewritten as E¯e˙(t)=Γ2ζ2(t).Remark 10.
From error dynamics (37), we can see that the new matrices A¯(t), A¯τ(t), C¯(t), C¯τ(t), B¯d(t), and D¯d(t) are known matrices, while the matrices L¯(t) contain two matrices L(t) and F(t) that have to be designed. Therefore, the proposed robust fault estimation observer design is converted to the problem of seeking the gain matrix L¯(t).Next, a fuzzy augmented fault estimation observer design method underH∞ performance is proposed to achieve robust fault estimation by following lemma under time-varying state delay.Lemma 11.
For the given positive scalarsτ1, τ2, τD, and γ, the error dynamic system (37) is admissible with ω(t)=0 while satisfying a prescribed H∞ performance (39), if there exist appropriately dimensional matrices P, Q1>0, Q2>0, S1>0, S2>0, R>0, and L¯(t) and free weighting matrices V=V1V2V3V4V5V6 and W=W1W2W3W4W5W6, such that the following inequalities hold:(41)E¯TP=PTE¯≥0,(42)Δ1t=Φtτ2-τ1VT∗-τ2-τ1R<0,(43)Δ2t=Φtτ2-τ1WT∗-τ2-τ1R<0,where (44)Φt=Φ1t+τ2-τ1Γ2TRΓ2,Φ1t=Φ11t-V1TEV1TE-W1TEW1TEΦ15tΦ16t∗Φ22Φ23W2TE-ETV4-ETV5-ETV6∗∗Φ33Φ34ETV5-ETW5ETV6-ETW6∗∗∗Φ44ETW5ETW6∗∗∗∗-ɛ1-τDQ10∗∗∗∗∗-γ2Iwith (45)Φ11t=sym⁡PTA¯t-L¯tC¯t+ɛQ1+Q2+I¯qI¯qT,Φ15t=PTA¯τt-L¯tC¯τt,Φ16t=PTL¯tD¯dt-B¯dt,I¯qT=0Iq,Φ22=S1-Q2-V2TE-ETV2,Ψ23=V2TE-W2TE-ETV3,Ψ33=S2-S1+symV3TE-W3TE,Ψ34=W3TE-ETV4-ETW4,Ψ44=-S2+W4TE+ETW4.Proof.
First, we show that the error dynamic system (37) with ω(t)=0 is regular and impulse-free. Since E¯ is singular and rank(E¯)=g+q, there always exist two nonsingular matrices G^∈R(n+q)×(n+q) and U^∈R(n+q)×(n+q) such that(46)G^E¯H^=Ig+q000.Accordingly, denote(47)G^A¯t-L¯tC¯tH^=G^A^tH^=A^11tA^12tA^21tA^22t,G^-TPH^=P^11P^12P^21P^22.From (41) and using the expressions in (46)-(47), it is easy to obtain that P^12=0. Since (42)-(43) hold, we have Φ(t)=Φ1(t)+(τ2-τ1)Γ2TRΓ2<0. Moreover, noting R>0 and by Schur complement, we can get Φ11(t)=sym(PT(A¯(t)-L¯(t)C¯(t)))+ɛQ1+Q2+I¯qI¯qT<0. Then, premultiplying and postmultiplying Φ11(t)<0 by H^T and H^, respectively, we have A^22T(t)P^22+P^22TA^22(t)<0, which implies that A^22(t) is nonsingular, and thus the pair (E¯,A^(t)) is regular and impulse-free. Hence, the error dynamic system (37) is regular and impulse-free for any time delay τ(t) satisfying (2) when inequalities (41)–(43) hold. Next, we will prove that the error dynamic system (37) is stable with H∞ performance. To this end, the Lyapunov-Krasovskii functional candidate is constructed as follows:(48)Vt=eTtE¯TPet+ɛ∫t-τtteTsQ1esds+∫t-τ1teTsQ2esds+∫t-τρt-τ1eTsS1esds+∫t-τ2t-τρeTsS2esds+∫-τ2-τ1∫t+θte˙TsE¯TRE¯e˙sdsdθ,where the unknown matrices P, Q1>0, Q2>0, S1>0, S2>0, and R>0 are to be determined. Then, the time derivatives of V(t) along the trajectories of the error dynamic system (37) satisfy (49)V˙t=eTtPTA¯t-L¯tC¯t+A¯t-L¯tC¯tTP+ɛQ1+Q2et+2eTtPTA¯τt-L¯tC¯τtet-τt+2eTtPTL¯tD¯dt-B¯dtωt-ɛ1-τ˙teTt-τtQ1et-τt+eTt-τ1S1-Q2et-τ1+eTt-τρS2-S1et-τρ+eTt-τ2-S2et-τ2+τ2-τ1E¯e˙tTRE¯e˙t-∫t-τ2t-τ1e˙TsE¯TRE¯e˙sds. Denoting β2Tt,s=ζ2T(t)(E¯e˙(s))T, and from Newton-Leibniz formula, a straightforward computation gives (50)
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s
. If (42)-(43) hold, one has V˙(t)+efT(t)ef(t)-γ2ωT(t)ω(t)<0. By noticing V(L)≥0 and V(0)=0 under zero initial conditions, we can conclude that (39) holds for all L>0 and any nonzero ω(t)∈L2[0,∞).
On the other hand, under conditions (42)-(43), by choosing the same Lyapunov function as (48) and following the similar line in the earlier deduction under conditions (42)-(43), then substitute Ai and Aτi by (A¯(t)-L¯(t)C¯(t)) and (A¯τ(t)-L¯(t)C¯τ(t)); we can easily obtain that the time derivative of V(t) along the solution of error dynamics (37) with ω(t)=0 satisfies V˙(et)<αet by using Theorem 7, which indicates the stability of system (37). This complete the proof.In the following, we will focus on the design of observer based on Lemma11 and provide a new sufficient condition for the existence of robust fault estimation observer for fuzzy singular time delay system (3).Theorem 12.
For the given positive scalarsτ1, τ2, τD, δ, and γ, the error dynamic system (37) is admissible with ω(t)=0 while satisfying a prescribed H∞ performance (39), if there exist appropriately dimensional matrices P, Q1>0, Q2>0, S1>0, S2>0, R>0, and Yi and free weighting matrices V=V1V2V3V4V5V6 and W=W1W2W3W4W5W6, such that the following inequalities hold:(51)E¯TP=PTE¯≥0,(52)Ξii<0i=1,2,…,r,(53)Ξij+Ξji≤01≤i<j≤r,(54)Πii<0i=1,2,…,r,(55)Πij+Πji≤01≤i<j≤r,where (56)Ξij=Φijτ2-τ1VTτ2-τ1Γ¯2ijT∗-τ2-τ1R0∗∗-2δE¯TP+δ2R,Πij=Φijτ2-τ1WTτ2-τ1Γ¯2ijT∗-τ2-τ1R0∗∗-2δE¯TP+δ2R, where (57)Φij=Φ11ij-V1TE¯V1TE¯-W1TE¯W1TE¯Φ15ijΦ16ij∗Φ22Φ23W2TE¯-E¯TV4-E¯TV5-E¯TV6∗∗Φ33Φ34E¯TV5-E¯TW5E¯TV6-E¯TW6∗∗∗Φ44E¯TW5E¯TW6∗∗∗∗-ɛ1-τDQ10∗∗∗∗∗-γ2I with (58)Φ11ij=sym⁡PTA¯i-YiC¯j+ɛQ1+Q2+I¯qI¯qT,Φ15ij=PTA¯τi-YiC¯τj,Φ16ij=YiD¯dj-PTB¯di,Φ22=S1-Q2-V2TE-ETV2,Ψ23=V2TE-W2TE-ETV3,Ψ33=S2-S1+sym⁡V3TE-W3TE,Ψ34=W3TE-ETV4-ETW4,Ψ44=-S2+W4TE+ETW4,I¯qT=0Iq,Γ¯2ij=E¯PTA¯i-YiC¯j000PTA¯τi-YiC¯τjYiD¯dj-PTB¯di. Then the observer gain matrices can be obtained as L¯i=LiFi=P-TYi.Proof.
For any scalarδ, it follows from the fact (δR-P)R-1(δR-P)≥0 that -PR-1P≤-2δP+δ2R. By Schur complement theorem, we can conclude that (42)-(43) hold if the following inequalities hold: (59)Φ1tτ2-τ1VTτ2-τ1Γ2TPE¯T∗-τ2-τ1R0∗∗-2δE¯PT+δ2R<0,Φ1tτ2-τ1WTτ2-τ1Γ2TPE¯T∗-τ2-τ1R0∗∗-2δE¯PT+δ2R<0, where Φ1(t) is defined in Lemma 11. Then, if (52)–(55) hold and with the changes of variables as Yi=PTLi, we have (60)Δ1t=∑i=1rμi2ξtΞii+∑i=1r∑i<jrμiξtμjξtΞij+Ξji<0,Δ2t=∑i=1rμi2ξtΠii+∑i=1r∑i<jrμiξtμjξtΠij+Πji<0 which imply that the error dynamics (37) are stable with ω(t)=0 while satisfying the prescribed H∞ performance (39) by Lemma 11. The proof is completed.Remark 13.
Theorem12 provides a criterion for designing H∞ robust fault estimation observer of fuzzy singular time delay systems, which guarantees the stability of the resulting dynamic error system with H∞ performance γ>0. As the delay term τ(t) is not simply enlarged, the proposed conditions are less conservative. Moreover, the proposed method is not only able to better depict the size and shape of the actuator fault but also able to estimate the sensor faults simultaneously.Remark 14.
Note that conditions (52)–(55) are LMIs. This indicates that the conditions (52)-(55) can be included as an optimization variable problems, which can be exploited to reduce the attenuation level bound. Then, the minimum attenuation level of H∞ performance can be obtained by the mincx function of Matlab toolbox. From the practical point of view, it is interesting to find an estimation law, which minimizes the disturbance rejection level γ for the error dynamic system. This can be done by solving a convex optimization problem P: min⁡ϑ subject to (51)–(55) with ϑ=γ2.Remark 15.
In dealing with time-varying faults, there may be a time delay between the fault estimation and the system fault. This phenomenon results from the influence of fault variation. Theoretically, the attenuation levelγmin can be minimized so that the fault estimation is insensitive to the fault variation. However, the cost is that the fault estimation becomes less robust to disturbance noise. Therefore, the attenuation level of γmin, the fault variation, and disturbance are a trade-off.


## 3.1. Delay-Dependent Stability

In this subsection, we suggest developing a delay-dependent stability condition for the nominal unforced fuzzy singular system of (7), which can be written as(13)Ex˙t=Atxt+Aτtxt-τt+Bdtdt,yt=Ctxt+Cτtxt-τt+Ddtdt,xt=ϕt,t∈-τ2,0.Theorem 7.
For the givenτ1, τ2, τD, the free fuzzy singular system (13) with d(t)=0 is admissible for any time-varying delay τ(t) satisfying (2), if there exists a nonsingular matrix P, symmetric positive-definite matrices Q1>0, Q2>0, S1>0, S2>0, and R>0, and any appropriately dimensioned matrices V=V1V2V3V4V5 and W=W1W2W3W4W5, such that the following set of inequalities hold:(14)ETP=PTE≥0,(15)Ψiτ2-τ1VT∗-τ2-τ1R<0,(16)Ψiτ2-τ1WT∗-τ2-τ1R<0,where (17)Ψi=Ψ1i+τ2-τ1Γ1TRΓ1,Ψ1i=Ψ11i-V1TEV1TE-W1TEW1TEPTAτi∗Ψ22Ψ23W2TE-ETV4-ETV5∗∗Ψ33Ψ34ETV5-ETW5∗∗∗Ψ44ETW5∗∗∗∗-ɛ1-τDQ1 with (18)Ψ11i=PTAi+AiTP+ɛQ1+Q2,Ψ22=S1-Q2-V2TE-ETV2,Ψ23=V2TE-W2TE-ETV3,Ψ33=S2-S1+sym⁡V3TE-W3TE,Ψ34=W3TE-ETV4-ETW4,Ψ44=-S2+W4TE+ETW4.Proof.
The proof of this theorem is divided into two parts. The first one is concerned with the regularity and the impulse-free characterizations, and the second one treats the stability property of system (13).
Since rank(E)=g≤n, there must exist two invertible matrices G∈Rn×n and H∈Rn×n such that(19)E~=GEH=Ig000.Similar to (19), we define(20)A~i=GAiH=GAi11Ai12Ai21Ai22H=A~i11A~i12A~i21A~i22,i=1,2,…,r,P~=G-TPH=G-TP11P12P21P22H=P~11P~12P~21P~22.Since Ψi<0 and Q1>0 and Q2>0 and R>0, we can formulate the following inequality easily:(21)Ϝ=AiTP+PTAi<0.Then, pre- and postmultiplying Ϝ<0 by HT and H, respectively, (21) yields(22)Ϝ~=A~iTP~+P~TA~i=Ϝ~11Ϝ~12∗A~i22TP~22+P~22TA~i22<0.Since Ϝ~11 and Ϝ~12 are irrelevant to the results of the following discussion, the real expression of these two variables is omitted here. From (22), it is easy to see that (23)A~i22TP~22+P~22TA~i22<0. Since μi(ξ(t))≥0 and ∑i=1rμi(ξ(t))=1, we have (24)∑i=1rμiξtA~i22TP~22+P~22TA~i22<0. This implies that ∑i=1rμi(ξ(t))A~i22 is nonsingular. Therefore, the unforced fuzzy singular system (13) is regular and impulse-free.
Next, we will show the stability of system (13). If conditions (15)-(16) hold, we have (25)∑i=1rμiPTAi+AiTP+ɛQ1+Q2PTAτi∗-ɛ1-τDQ1<0. Premultiplying and postmultiplying the preceding inequality by (26)0I00000I and its transposes, respectively, we obtain (27)sym⁡P22T∑i=1rμiAi22+ɛQ122+Q222P22T∑i=1rμiAτi22∗-ɛ1-τDQ122<0. which implies that(28)
sym
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122
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∗
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122
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.It follows from (1,1)-block of (28) that Ai22 is invertible. Then, premultiplying and postmultiplying the preceding inequality by (29)
-
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0 and its transpose, respectively, yield (30)∑i=1rμiAi22-1∑i=1rμiAτi22TQ122∑i=1rμiAi22-1∑i=1rμiAτi22-1-τDQ122<0 which shows that ρ((∑i=1rμiAi22)-1(∑i=1rμiAτi22))<1 holds for all allowable μi.
Now, let us choose the following Lyapunov-Krasovskii function as(31)Vt=V1t+V2t+V3t, where(32)V1t=xTtETPxt,V2t=ɛ∫t-τttxTsQ1xsds+∫t-τ1txTsQ2xsds+∫t-τρt-τ1xTsS1xsds+∫t-τ2t-τρxTsS2xsds,V3t=∫-τ2-τ1∫t+θtx˙TsETREx˙sdsdθ, where the unknown matrices P, Q1>0, Q2>0, S1>0, S2>0, and R>0 are to be determined, and τρ=τ1+ρδ, δ=τ2-τ1, and 0<ρ<1. Then, the time derivatives of V(t) along the trajectories of the dynamic systems (13) satisfy (33)V˙1t=xTtPTAt+ATtPxt+2xTtPTAτtxt-τt,V˙2t=ɛxTtQ1xt-ɛ1-τ˙txTt-τtQ1xt-τt+xTtQ2xt-xTt-τ1Q2xt-τ1+xTt-τ1S1xt-τ1-xTt-τρS1xt-τρ+xTt-τρS2xt-τρ-xTt-τ2S2xt-τ2,V˙3t=τ2-τ1Ex˙tTREx˙t-∫t-τ2t-τ1x˙TsETREx˙sds=τ2-τ1Ex˙tTREx˙t-∫t-τρt-τ1x˙TsETREx˙sds-∫t-τ2t-τρx˙TsETREx˙sds. Denoting Γ1=A(t)000Aτ(t),  ζ1T(t)=xT(t)xT(t-τ1)xT(t-τρ)xT(t-τ2)xT(t-τ(t)) and β1Tt,s=ζ1T(t)(Ex˙(s))T, and from Newton-Leibniz formula, we can easily obtain that(34)-2ζ1TtVTExt-τ1-Ext-τρ-∫t-τρt-τ1Ex˙sds=0,-2ζ1TtWTExt-τρ-Ext-τ2-∫t-τ2t-τρEx˙sds=0. Therefore, a straightforward computation gives (35)
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1
t
,
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d
s
. Then, if conditions (15)-(16) hold, there exists α>0 such that V˙(xt)<αxt. By Lemma 3, we conclude that the unforced fuzzy singular system (13) is stable. This completes the proof.Remark 8.
In some existing literature, for example, [22, 23], some delay-dependent criteria are given in terms of LMIs to guarantee that the fuzzy singular system is admissible by using LKF approach and integral inequality, such as Lemma  2 in [22] and Lemma  2.3 in [23]. However, in the proof of our result of Theorem 7, we use one identical equality to estimate the upper bound of the derivative of V(t) without any model transformation. Moreover, it is interesting to mention that this study presents criteria based on the free weighting matrix method, in which the bounding techniques on some cross product terms are not involved [11, 23]. The major feature of this method is to reduce the conservatism engendered by the system transformations and the bounding techniques.Remark 9.
For time delay systems, the Lyapunov functional candidate always involves the integral term∫-τ0∫t+θtx˙TsRx˙sdsdθ, and the derivative of it was estimated as τx˙T(t)Rx˙(t)-∫t-τtx˙T(s)Rx˙(s)ds. However, the term ∫t-τtx˙T(s)Rx˙(s)ds was ignored [25–27], or some useful negative integral term was lost; see, for example, [28, 29]. Instead, in this paper all those terms ∫t-τρt-τ1x˙T(s)Rx˙(s)dsand ∫t-τ2t-τρx˙T(s)Rx˙(s)ds, which contain a great amount of useful information about systems, are preserved. Therefore, it is obvious to see that this method will lead to less conservatives than the existing ones in [25–29]. Furthermore, the introduction of parameter ɛ(ɛ≥0) indicates that Lemma 3 can be suitable for time-varying delay τ(t) being unknown or not differentiable; that is, in the case of time-varying delay τ(t) not differentiable, one can set ɛ=0.In order to estimate system faults, the following fault estimation observer is constructed:(36)Ex^˙t=Atx^t+Aτtx^t-τt+Btut+Bftf^t-Lty^t-yt,y^t=Ctx^t+Cτtx^t-τt+Dtut+Dftf^t,f^˙t=-Fty^t-yt,where x^(t)∈Rn is the observer state, y^(t)∈Rl is the observer output, and f^(t)∈Rq is an estimate of fault f(t). The objective is to design the appropriate dimension gain matrices L(t)∈Rn×l and F(t)∈Rq×l and estimate the fault despite the presence of the disturbance and state delay, where L(t)=∑i=1rμi(ξ(t))Li and F(t)=∑i=1rμi(ξ(t))Fi. Let us define ex(t)=x^(t)-x(t), ey(t)=y^(t)-y(t), ef(t)=f^(t)-f(t), and eT(t)=[exT(t),efT(t)], ωT(t)=[dT(t),f˙T(t)]; then the error dynamic systems are deduced from (7) and (36) as follows:(37)E¯e˙t=A¯t-L¯tC¯tet+A¯τt-L¯tC¯τtet-τt+L¯tD¯dt-B¯dtωt,eyt=C¯tet+C¯τtet-τt-D¯dtωt,where (38)E¯=E00Iq,A¯t=AtBft00,A¯τt=Aτt000,B¯dt=Bdt00Iq,L¯t=LtFt,C¯t=CtDft,C¯τt=Cτt0,D¯dt=Ddt0. Therefore, H∞ robust fault estimation observer design problem to be addressed in this paper can be formulated as follows: (i) The error dynamic system (37) with ω(t)=0 is admissible for any time delay satisfying (2); (ii) for a given scalar γ, the following H∞ performance is satisfied:(39)∫0Left2dt≤γ2∫0Lωt2dtfor all L>0 and ω(t)∈L2[0,∞) under zero initial conditions.For simplicity, we introduce the following vectors:(40)ζ2Tt=eTteTt-τ1eTt-τρeTt-τ2eTt-τtωTt,Γ2=A¯t-L¯tC¯t000A¯τt-L¯tC¯τtL¯tD¯dt-B¯dt. Then, the state of error dynamics (37) can be rewritten as E¯e˙(t)=Γ2ζ2(t).Remark 10.
From error dynamics (37), we can see that the new matrices A¯(t), A¯τ(t), C¯(t), C¯τ(t), B¯d(t), and D¯d(t) are known matrices, while the matrices L¯(t) contain two matrices L(t) and F(t) that have to be designed. Therefore, the proposed robust fault estimation observer design is converted to the problem of seeking the gain matrix L¯(t).Next, a fuzzy augmented fault estimation observer design method underH∞ performance is proposed to achieve robust fault estimation by following lemma under time-varying state delay.Lemma 11.
For the given positive scalarsτ1, τ2, τD, and γ, the error dynamic system (37) is admissible with ω(t)=0 while satisfying a prescribed H∞ performance (39), if there exist appropriately dimensional matrices P, Q1>0, Q2>0, S1>0, S2>0, R>0, and L¯(t) and free weighting matrices V=V1V2V3V4V5V6 and W=W1W2W3W4W5W6, such that the following inequalities hold:(41)E¯TP=PTE¯≥0,(42)Δ1t=Φtτ2-τ1VT∗-τ2-τ1R<0,(43)Δ2t=Φtτ2-τ1WT∗-τ2-τ1R<0,where (44)Φt=Φ1t+τ2-τ1Γ2TRΓ2,Φ1t=Φ11t-V1TEV1TE-W1TEW1TEΦ15tΦ16t∗Φ22Φ23W2TE-ETV4-ETV5-ETV6∗∗Φ33Φ34ETV5-ETW5ETV6-ETW6∗∗∗Φ44ETW5ETW6∗∗∗∗-ɛ1-τDQ10∗∗∗∗∗-γ2Iwith (45)Φ11t=sym⁡PTA¯t-L¯tC¯t+ɛQ1+Q2+I¯qI¯qT,Φ15t=PTA¯τt-L¯tC¯τt,Φ16t=PTL¯tD¯dt-B¯dt,I¯qT=0Iq,Φ22=S1-Q2-V2TE-ETV2,Ψ23=V2TE-W2TE-ETV3,Ψ33=S2-S1+symV3TE-W3TE,Ψ34=W3TE-ETV4-ETW4,Ψ44=-S2+W4TE+ETW4.Proof.
First, we show that the error dynamic system (37) with ω(t)=0 is regular and impulse-free. Since E¯ is singular and rank(E¯)=g+q, there always exist two nonsingular matrices G^∈R(n+q)×(n+q) and U^∈R(n+q)×(n+q) such that(46)G^E¯H^=Ig+q000.Accordingly, denote(47)G^A¯t-L¯tC¯tH^=G^A^tH^=A^11tA^12tA^21tA^22t,G^-TPH^=P^11P^12P^21P^22.From (41) and using the expressions in (46)-(47), it is easy to obtain that P^12=0. Since (42)-(43) hold, we have Φ(t)=Φ1(t)+(τ2-τ1)Γ2TRΓ2<0. Moreover, noting R>0 and by Schur complement, we can get Φ11(t)=sym(PT(A¯(t)-L¯(t)C¯(t)))+ɛQ1+Q2+I¯qI¯qT<0. Then, premultiplying and postmultiplying Φ11(t)<0 by H^T and H^, respectively, we have A^22T(t)P^22+P^22TA^22(t)<0, which implies that A^22(t) is nonsingular, and thus the pair (E¯,A^(t)) is regular and impulse-free. Hence, the error dynamic system (37) is regular and impulse-free for any time delay τ(t) satisfying (2) when inequalities (41)–(43) hold. Next, we will prove that the error dynamic system (37) is stable with H∞ performance. To this end, the Lyapunov-Krasovskii functional candidate is constructed as follows:(48)Vt=eTtE¯TPet+ɛ∫t-τtteTsQ1esds+∫t-τ1teTsQ2esds+∫t-τρt-τ1eTsS1esds+∫t-τ2t-τρeTsS2esds+∫-τ2-τ1∫t+θte˙TsE¯TRE¯e˙sdsdθ,where the unknown matrices P, Q1>0, Q2>0, S1>0, S2>0, and R>0 are to be determined. Then, the time derivatives of V(t) along the trajectories of the error dynamic system (37) satisfy (49)V˙t=eTtPTA¯t-L¯tC¯t+A¯t-L¯tC¯tTP+ɛQ1+Q2et+2eTtPTA¯τt-L¯tC¯τtet-τt+2eTtPTL¯tD¯dt-B¯dtωt-ɛ1-τ˙teTt-τtQ1et-τt+eTt-τ1S1-Q2et-τ1+eTt-τρS2-S1et-τρ+eTt-τ2-S2et-τ2+τ2-τ1E¯e˙tTRE¯e˙t-∫t-τ2t-τ1e˙TsE¯TRE¯e˙sds. Denoting β2Tt,s=ζ2T(t)(E¯e˙(s))T, and from Newton-Leibniz formula, a straightforward computation gives (50)
V
˙
t
+
e
f
T
t
e
f
t
-
γ
2
ω
T
t
ω
t
=
V
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t
+
e
T
t
I
¯
q
I
¯
q
T
e
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-
γ
2
ω
T
t
ω
t
=
e
T
t
P
T
A
¯
t
-
L
¯
t
C
¯
t
+
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¯
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¯
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+
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+
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+
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+
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+
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1
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2
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1
+
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2
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1
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+
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t
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2
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+
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t
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1
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2
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t
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e
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s
d
s
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2
ζ
2
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t
W
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E
¯
0
0
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-
I
0
0
ζ
2
t
+
∫
t
-
τ
2
t
-
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ρ
2
ζ
2
T
t
W
T
E
¯
e
˙
s
d
s
≤
1
τ
2
-
τ
1
∫
t
-
τ
ρ
t
-
τ
1
ζ
2
T
t
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t
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2
t
+
2
τ
2
-
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1
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2
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t
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e
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s
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2
-
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1
e
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e
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d
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+
1
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1
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t
-
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2
t
-
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ρ
ζ
2
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t
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t
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2
t
+
2
τ
2
-
τ
1
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2
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t
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T
E
¯
e
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s
-
τ
2
-
τ
1
e
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s
E
¯
T
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E
¯
e
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s
d
s
=
1
τ
2
-
τ
1
∫
t
-
τ
ρ
t
-
τ
1
β
2
T
t
,
s
Φ
t
τ
2
-
τ
1
V
T
∗
-
τ
2
-
τ
1
R
β
2
t
,
s
d
s
+
1
τ
2
-
τ
1
∫
t
-
τ
2
t
-
τ
ρ
β
2
T
t
,
s
Φ
t
τ
2
-
τ
1
W
T
∗
-
τ
2
-
τ
1
R
β
2
t
,
s
d
s
. If (42)-(43) hold, one has V˙(t)+efT(t)ef(t)-γ2ωT(t)ω(t)<0. By noticing V(L)≥0 and V(0)=0 under zero initial conditions, we can conclude that (39) holds for all L>0 and any nonzero ω(t)∈L2[0,∞).
On the other hand, under conditions (42)-(43), by choosing the same Lyapunov function as (48) and following the similar line in the earlier deduction under conditions (42)-(43), then substitute Ai and Aτi by (A¯(t)-L¯(t)C¯(t)) and (A¯τ(t)-L¯(t)C¯τ(t)); we can easily obtain that the time derivative of V(t) along the solution of error dynamics (37) with ω(t)=0 satisfies V˙(et)<αet by using Theorem 7, which indicates the stability of system (37). This complete the proof.In the following, we will focus on the design of observer based on Lemma11 and provide a new sufficient condition for the existence of robust fault estimation observer for fuzzy singular time delay system (3).Theorem 12.
For the given positive scalarsτ1, τ2, τD, δ, and γ, the error dynamic system (37) is admissible with ω(t)=0 while satisfying a prescribed H∞ performance (39), if there exist appropriately dimensional matrices P, Q1>0, Q2>0, S1>0, S2>0, R>0, and Yi and free weighting matrices V=V1V2V3V4V5V6 and W=W1W2W3W4W5W6, such that the following inequalities hold:(51)E¯TP=PTE¯≥0,(52)Ξii<0i=1,2,…,r,(53)Ξij+Ξji≤01≤i<j≤r,(54)Πii<0i=1,2,…,r,(55)Πij+Πji≤01≤i<j≤r,where (56)Ξij=Φijτ2-τ1VTτ2-τ1Γ¯2ijT∗-τ2-τ1R0∗∗-2δE¯TP+δ2R,Πij=Φijτ2-τ1WTτ2-τ1Γ¯2ijT∗-τ2-τ1R0∗∗-2δE¯TP+δ2R, where (57)Φij=Φ11ij-V1TE¯V1TE¯-W1TE¯W1TE¯Φ15ijΦ16ij∗Φ22Φ23W2TE¯-E¯TV4-E¯TV5-E¯TV6∗∗Φ33Φ34E¯TV5-E¯TW5E¯TV6-E¯TW6∗∗∗Φ44E¯TW5E¯TW6∗∗∗∗-ɛ1-τDQ10∗∗∗∗∗-γ2I with (58)Φ11ij=sym⁡PTA¯i-YiC¯j+ɛQ1+Q2+I¯qI¯qT,Φ15ij=PTA¯τi-YiC¯τj,Φ16ij=YiD¯dj-PTB¯di,Φ22=S1-Q2-V2TE-ETV2,Ψ23=V2TE-W2TE-ETV3,Ψ33=S2-S1+sym⁡V3TE-W3TE,Ψ34=W3TE-ETV4-ETW4,Ψ44=-S2+W4TE+ETW4,I¯qT=0Iq,Γ¯2ij=E¯PTA¯i-YiC¯j000PTA¯τi-YiC¯τjYiD¯dj-PTB¯di. Then the observer gain matrices can be obtained as L¯i=LiFi=P-TYi.Proof.
For any scalarδ, it follows from the fact (δR-P)R-1(δR-P)≥0 that -PR-1P≤-2δP+δ2R. By Schur complement theorem, we can conclude that (42)-(43) hold if the following inequalities hold: (59)Φ1tτ2-τ1VTτ2-τ1Γ2TPE¯T∗-τ2-τ1R0∗∗-2δE¯PT+δ2R<0,Φ1tτ2-τ1WTτ2-τ1Γ2TPE¯T∗-τ2-τ1R0∗∗-2δE¯PT+δ2R<0, where Φ1(t) is defined in Lemma 11. Then, if (52)–(55) hold and with the changes of variables as Yi=PTLi, we have (60)Δ1t=∑i=1rμi2ξtΞii+∑i=1r∑i<jrμiξtμjξtΞij+Ξji<0,Δ2t=∑i=1rμi2ξtΠii+∑i=1r∑i<jrμiξtμjξtΠij+Πji<0 which imply that the error dynamics (37) are stable with ω(t)=0 while satisfying the prescribed H∞ performance (39) by Lemma 11. The proof is completed.Remark 13.
Theorem12 provides a criterion for designing H∞ robust fault estimation observer of fuzzy singular time delay systems, which guarantees the stability of the resulting dynamic error system with H∞ performance γ>0. As the delay term τ(t) is not simply enlarged, the proposed conditions are less conservative. Moreover, the proposed method is not only able to better depict the size and shape of the actuator fault but also able to estimate the sensor faults simultaneously.Remark 14.
Note that conditions (52)–(55) are LMIs. This indicates that the conditions (52)-(55) can be included as an optimization variable problems, which can be exploited to reduce the attenuation level bound. Then, the minimum attenuation level of H∞ performance can be obtained by the mincx function of Matlab toolbox. From the practical point of view, it is interesting to find an estimation law, which minimizes the disturbance rejection level γ for the error dynamic system. This can be done by solving a convex optimization problem P: min⁡ϑ subject to (51)–(55) with ϑ=γ2.Remark 15.
In dealing with time-varying faults, there may be a time delay between the fault estimation and the system fault. This phenomenon results from the influence of fault variation. Theoretically, the attenuation levelγmin can be minimized so that the fault estimation is insensitive to the fault variation. However, the cost is that the fault estimation becomes less robust to disturbance noise. Therefore, the attenuation level of γmin, the fault variation, and disturbance are a trade-off.


## 4. Numerical Examples

In this section, three examples are given to show the effectiveness of our results. All the numerical results are calculated via the Yalmip toolbox of Matlab.Example 1.
Consider a continuous fuzzy singular system composed of two rules and the following system matrices [22, 23]: (61)E=1000010000100000,A1=-3000.20-40.1000-0.100.10.1-0.2-0.2,A2=-200-0.20-2.5-0.100-0.2-0.300.10.1-0.2-0.2,Aτ1=-0.50000-10000.1-0.200000,Aτ2=-0.50000-10000.1-0.500000. Assume that the delay τ(t) satisfies (2) and set ρ=0.5. The obtained results are listed in Table 1, where N(m) stands for the total number of decision variables. Table 1 tabulates a comparison of the maximum allowable upper delay bound τ2 for a prescribed τD. It can be seen from the table that our results are marked better than those obtained by the method in [22, 23]. Moreover, it is worth mentioning that the method proposed in this paper uses fewer number of LMIs scalar variables and fewer number of LMIs for stability computation; thus our method is more computationally efficient for improving the upper bound of delay; the stability criterion we derived is less conservative than those reported in the aforementioned papers.Table 1
Allowable upper bound ofτ2 for various τD in Example 1.
τ
D
0.1
0.35
0.6
0.85
0.9
0.95
N
(
m
)
Theorem1 in [22]
3.3623
2.9810
2.6010
1.8330
1.3080
—
24
Theorem3.1 in [23]
3.3685
3.1560
3.1510
3.0760
2.6750
2.0780
19
Theorem7
4.1230
3.9891
3.7104
3.5247
3.4450
3.3702
15Example 2.
Consider the following singular time delay system:(62)1000x˙t=0.63410.5413-0.6121-1.1210xt+-0.450000-0.1210xt-h. To compare with the existing results, we assume that τD=0 and ρ=0.5. Table 2 lists the comparison results on the maximum allowed time delay τ2 via the methods in [10, 20, 22–24] and Theorem 7 in this paper. From the comparison result we can see that the stability criterion we derived by using free weighting matrix approach in this work is less conservative than those reported in [10, 20, 22–24].Table 2
Comparison of maximum allowed delaysτ2 in Example 2.
[10]
[20]
[24]
[22]
[23]
Theorem7
—
2.1328
2.1372
2.1372
2.3393
3.1324Example 3.
Consider the following fuzzy singular system with time-varying delay:(63)Ex˙t=∑i=12μiξtAixt+Aτixt-τt+Biut+Bfift+Bdidt,yt=∑i=12μiξtCixt+Cτixt-τt+Dfift+Ddidt, where (64)E=100010000,A1=-0.601-1-0.65-0.50.20.74-4-1,A2=-0.601-0.8-0.65-0.500.40.82-6.37-1,Aτ1=0.070.02-0.01-0.0900.020.070.050,Aτ2=0.070.12-0.01-0.0800.030.120.040,Bi=Bfi=0.700,Ddi=0.2,i=1,2,Bd1=Bd2=0.10.10.1,C1=C2=-0.20.5-0.15,Ct1=Ct2=-0.020.05-0.15. Here, we consider the case where δ=2 and the time-varying delay is given as τt=0.3+0.2sin⁡(t), and a straightforward calculation gives τ1=0.1, τ2=0.5, and τD=0.2. By solving the conditions in (51)–(55), we obtain that the achieved γmin=1.3140, and the feasible solution is (due to space consideration, we do not list all the matrices here) (65)P=10.216811.61310.0000-4.284011.613132.76650.0000-2.85470.00000.00001.48420.0000-4.2840-2.85470.00002.1990,Y1=4.095750.1306-7.9107-3.6154,Y2=4.294348.5194-10.5807-3.7476. The associate fault estimation observer gains in (36) are (66)L1=-20.32175.7954-5.3299,F1=-33.7106,L2=-19.81035.6276-7.1289,F2=-32.9922.
According to Theorem12, we can consider different δ to find the minimum index γ for the given τD=0.2; see [25, 28] for more details. The corresponding results are summarized in Table 3.
In order to illustrate the performance of robust fault estimation observer in dealing with fuzzy singular systems with time delay, first, an abrupt fault is simulated. It is assumed that the abrupt faultf(t) is created as(67)ft=00≤t<10-1010≤t≤30. For simulation purpose, we choose the membership functions for Rules 1 and 2 to be μ1(ξ(t))=1/(1+exp(x1(t)+0.5)) and μ2(ξ(t))=1-μ1(ξ(t)) with the initial state condition as x0=-1.571-1.3561.279T while the initial estimate is x^(0)=000T. Meanwhile, it is assumed that d(t) is band-limited white noise with power 0.1 and sampling time 0.01s. Then, the fault estimation result of the robust fault estimation observer is depicted in Figure 1. Therein, the fault is depicted by red dash-and-dot line, and the fault estimation is represented by the blue solid one. As shown in Figure 1, the robust fault estimation observer is insensitive to the model disturbance. Moreover, although there is estimation error, the fault estimate can quickly track the fault. This illustrates the fast convergence rate of the fault estimation observer in the face of initial estimation error. The simulation results shown in Figure 1 obviously illustrate that the proposed fault estimation has a good performance to estimate fault, and the error dynamic system is also stable in Figure 2. To illustrate the performance of robust fault estimation observer design method, the following time-varying fault is further considered: (68)f1t=00≤t<10101-e-t-510≤t≤20,f2t=00≤t<100.5sin⁡0.8t-510≤t≤20.
In this situation, the fault estimation result is depicted in Figures3 and 5. It can be seen from Figures 3 and 5 that the fault is estimated with satisfactory accuracy and rapidity. The state of the error dynamic system is also stable in Figures 4 and 6. In [13], actuator fault estimation observer is designed for discrete-time linear parameter-varying descriptor systems, but the systems with time-varying delay case are not considered. In [9], pole assignment is used to ensure the fault estimation convergence speed while our method utilizes H∞ technique to attenuate the effect of fault variation. From Figures 3 and 5, it can be seen that these methods have similar fault convergence speed. Nevertheless, [9] only deals with regular systems not with time delay. Moreover, our method considers the fuzzy singular system with actuator and sensor faults simultaneously. Therefore, the proposed method is more general than that in [9, 13].Table 3
Minimum indexγ for various δ in Example 3 with τD=0.2.
δ
0.5
1
2
5
10
τ
=
0.5
1.6322
1.4559
1.3732
1.3465
1.5532Figure 1
Fault estimation result of the robust fault estimation observer in abrupt fault casef(t).Figure 2
Response curves of error dynamicse1(t), e2(t), e3(t), and ef(t) in abrupt fault case f(t).Figure 3
Fault estimation result of the robust fault estimation observer in faultf1(t).Figure 4
Response curves of error dynamicse1(t), e2(t), e3(t), and ef(t) in fault f1(t).Figure 5
Fault estimation result of the robust fault estimation observer in faultf2(t).Figure 6
Response curves of error dynamicse1(t), e2(t), e3(t), and ef(t) in fault f2(t).


## 5. Conclusion

In this paper, the robust fault estimation problem for T-S fuzzy singular systems with time-varying delays is considered. By considering the fault as an auxiliary disturbance vector, based on the Lyapunov theorem and improved delay partitioning method with free weighting matrix approach, we give some less conservative criteria, which guarantee that the considered system is regular, impulse-free, and stable, while limiting the influence of disturbance despite the presence of actuator and sensor faults simultaneously. This paper proposes a novel fault estimation observer and presents an LMI-based design method for the fuzzy singular system. Finally, some numerical examples are used to demonstrate the effectiveness and performance of the proposed method.



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*Source: 6305901-2016-03-07.xml*