# Robust Fast Adaptive Fault Estimation for T-S Fuzzy Markovian Jumping Systems with Mode-Dependent Time-Varying State Delays

Authors: Chao Sun; Shengjuan Huang; Libing Wu; Suhuan Yi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper studies the problem of actuator fault estimation for a class of T-S fuzzy Markovian jumping systems, which is subject to mode-dependent interval time-varying delays and norm-bounded external disturbance. Based on the given fast adaptive estimation algorithm and by employing a novel Lyapunov–Krasovskii function candidate, a robust fault estimation scheme is proposed to estimate faults whose derivative is bounded. With this improved method, the proposed fault estimator minimizes the effect of disturbance on the estimation error and reduces the conservatism of systems stability results simultaneously. To be specific, an improved mode-dependent criterion for the existence of the fault estimation observer is established to guarantee the error dynamic system to be stochastically stable with a prescribedH∞ performance and reduce the conservatism of designing procedure. Finally, three numerical examples are given to show the effectiveness of the proposed method.


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## Body

## 1. Introduction

As an important class of stochastic systems, Markovian jump systems (MJSs) have been widely concerned over the past several decades. This kind of system model can be utilized to represent many types of physical systems in applications, such as electric power systems [1], manufacturing systems [2], and circuit systems [3]. The study of MJSs has been of both theoretical and practical importance, and many results have been reported in the literatures [4–6]. However, the existing results are mainly about the linear MJSs; little attention has been paid to nonlinear MJSs. Recent years have witnessed the growing interests of the Takagi–Sugeno (T-S) fuzzy model [7]. Extensive research suggests that the T-S fuzzy model can provide a general framework to approximate the nonlinear system, such as stochastic systems and hybrid systems. Therefore, it is natural that the nonlinear MJSs are represented by T-S fuzzy modes which can be described by the fuzzy IF-THEN rules. So far, under the framework of fuzzy logic control strategy, considerable attention has been paid to the stability analysis and controller synthesis for MJSs, such as robust filtering [8, 9] and H∞ control [10–13]. Recently, it is worth mentioning that, in [12], with the aid of a mode-dependent Lyapunov–Krasovskii functional and the Abel lemma-based finite sum inequality, the finite-time bounded control problem has been solved for a class of discrete-time T-S fuzzy Markov jump systems with memory state feedback controller. By exploiting Lyapunov stability theory and Wirtinger-based integral inequalities, literature [13] deals with the robust trajectory tracking control problem for T-S fuzzy Markovian jump systems with time-varying delay, uncertainties, and disturbances.On another research front line, due to the more increasing demand for reliability and safety in industrial control processes, the issues of fault detection and isolation (FDI), fault-tolerant control algorithms, and their applications to a wide range of industrial processes have been the subjects of intensive investigation in the recent years. For instance, by using adaptive backstepping technique, the work of Wang et al. [14] proposes a fault-tolerant compensation control approach against nonlinearity and simultaneous additive and multiplicative actuator faults. In [15], the fault-tolerant control problem against sensor faults of a time-delay Markov jump systems with Ito^ stochastic and output disturbances has been investigated by using the sliding mode observer. However, in order to achieve the better control effect in practice engineering, it is generally difficult to obtain the exact information of the size of system fault from an FDI strategy because of the existence of the unexpected model uncertainties, disturbances, perturbations, and noises that may also occur in the fault systems [16]. For increased efficiency, the problem of fault estimation (FE) has stirred renewed research interest; accordingly, a variety of important results have been developed in the literatures [17–23]. Compared with FDI, FE can give the size and shape of the fault and can thus automatically perform the required fault detection. However, FE is a more challenging task because it requires an estimation of the location after the alarm has been set, and the size of the fault should be made.Recently, many fault estimation approaches based on robust or adaptive observers have been developed in the literature. The problem of fault estimation and fault-tolerant control for stochastic systems and nonlinear descriptor systems were studied in [24, 25]. Also, Liu et al. [26] investigate the fault-tolerant control for nonlinear Markovian jump systems based on the sliding mode observer technique. In [27], the problem of fault estimation for a class of linear parameter varying Markovian jump systems with the actuator and sensor faults was studied based on the Cholesky decomposition technique. Based on the Lyapunov–Razumikhin method, an iterative algorithm is proposed to solve the robust fault estimation for a class of uncertain networked control systems in [28]. The literature [29] presents a novel method to address a proportional integral observer design for the actuator and sensor faults’ estimation based on the T-S fuzzy model with unmeasurable premise variables. Based on a discrete mean-value convergence lemma, an improved iterative observer for fault estimation of a class of discrete-time T-S fuzzy systems is discussed in [30]. By introducing the reset element in the observer, Pourdehi and Karimaghaee [31] investigated the problem of actuator fault estimation and FTC for a class of T-S fuzzy nonlinear time-delay systems. You et al. [19] proposed a proportional multiple integral observer to achieve simultaneous estimation of system states, time-varying actuator, and sensor faults for T-S fuzzy systems.It is well known that the presence of time delay often degrades system performances and may even lead to instability; many criteria have been advanced for checking the stability of MJSs. Because the delay-dependent results usually have less conservatism in contrast to the delay-independent ones, hence, some improved approaches have been proposed for deriving the delay-dependent results on linear MJSs with time delays, such as delay decomposition method [32, 33] and free weighting method [34, 35]. Based on the abovementioned methods, the delay-dependent fault estimation and parity space-based fault estimation schemes are studied in [36, 37]. The result in [17] is developed to research the fault estimation problem based on the condition that the time delay is constant, but the external disturbance is not considered, which inevitably brings some conservativeness. Chen et al. [18] investigate the problem of FE for a class of Markovian jump systems with time-varying delay, but the delay mode is the same as that for jump in other system states. In [38], robust H∞ control for a class of uncertain stochastic Markovian jump systems with interval and distributed time-varying delays is researched, but the problem of FE is not considered. It should be mentioned that although there are many results regarding the fault estimation for time delay T-S fuzzy jump systems, all such works have considered only time delay that share the same mode with system matrices, and the time delay have a fixed type under different fuzzy rules. Different from the abovementioned delays, in practical systems, with the change of time t, the state delays respond to changes in system modes. Therefore, in order to ensure a stable and efficient work system, the mode-dependent time-varying state delay is an important subject to be researched. In summary, despite these efforts to deal with time delay MJSs, to the best of our knowledge, there are only few conclusions of robust fault estimation for T-S fuzzy MJSs with actuator fault, external distu rbance, and mode-dependent interval time-varying delay, simultaneously, which has not been fully investigated. This motivates the present study.In this paper, we are concerned with the problem of robust fault estimation for a class of T-S fuzzy MJSs with external disturbances and mode-dependent time-delay parameters. Our aim is to design a mode-dependent fuzzy FE observer such that the error dynamic is stochastically stable with a prescribedH∞ performance of disturbance attenuation. Meanwhile, the proposed adaptive fault estimation algorithm can enhance the rapidity and accuracy performance of fault estimation. The main contributions of this paper are outlined as follows:(i)
Different from the existing results in [17, 20, 22], the mode-dependent time-varying delay is considered, which may not be the same as that for jump in system states. Then, a fast fault estimation observer is proposed, which can not only provide estimations of the actuator fault but also guarantee the good performance when the system and time delay change in different modes.(ii)
Both lower-upper bounds and the integral term of mode-dependent delays are considered in the construction of the LKF function. Moreover, a derivative term is added on the basis of fault estimation equation in the estimation algorithm. The improved mode-dependent fault estimator existence criteria are obtained than existing results in [17, 18, 22].(iii)
The robust fault estimation criterion of the error dynamic system can be readily obtained by means of LMIs. Compared with the existing results in [22], the proposed method has the advantage of few variables and wider application. Three numerical examples are provided to show the effectiveness of the proposed method.The rest of this paper is organized as follows. The system description and problem formulations are presented inSection 2. Section 3 presents the main results on the robust FE observer design scheme. In Section 4, numerical examples are presented to demonstrate the effectiveness 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>0X≥0 means that matrix X is real symmetric positive definite (positive semidefinite), and · is the spectral norm. The symbol ″∗″ stands for matrix block induced by symmetry. If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations.


## 2. Problem Formulation

Let us consider a class of nonlinear MJSs with mode-dependent time-varying state delays described by the following T-S fuzzy models over the probability spaceΩ,M,Pr, where Ω is the sample space, M is the algebra of events, and Pr is the probability measure defined on M.Plant Rulei: IF ξ1t is Mi1, and …, and ξpt is Mip, THEN(1)x˙t=Airtxt+Adirtxt−dt,lt+Birtut+Bfirtft+Dirtωt,yt=Cirtxt,xt=αt,rt=βt,lt=γt,∀t∈−h2,0,i=1,2,…,s,where xt∈ℝn is the state of the nonlinear model, yt∈ℝl stands for the measured output, ut∈ℝw denotes the controlled input, ωt∈ℝm is the exogenous disturbance that belongs to L20,∞, ft∈ℝq represents the possible actuator fault, Miji=1,2,…,s,j=1,2,…,p are fuzzy sets, ξt=ξ1t,…,ξptT is the available premise variable vector that does not depend on the input variables ut, Airt, Adirt, Birt, Bfirt, Cirt, and Dirt are known mode-dependent constant matrices with appropriate dimensions, and dt,lt is the mode-dependent time-varying delay satisfying 0≤h1l≤dt,lt≤h2l<∞ and d˙t,lt≤h3l<1.Here, let the random process {rt, t≥0} and {lt, t≥0} be continuous-time discrete-state Markov stochastic processes taking values on a finite set Λ1=1,2,…,N1 and Λ2=1,2,…,N2 with transition rate matrix Π1=πrk, r,k∈Λ1, and Π2=πlm, l,m∈Λ2, respectively. Meanwhile, we also assume that rt,t≥0 and lt,t≥0 are mutually independent. The transition probabilities from mode r at time t to mode k at time t+Δt of the random Markov process rt,t≥0 and from mode l at time t to mode m at time t+Δt of the random Markov process lt,t≥0 are(2)Pr=Prrt+Δt=k|rt=r=πrkΔt+oΔt,ifr≠k,1+πrrΔt+oΔt,ifr=k,Pl=Pllt+Δt=m|lt=l=πlmΔt+oΔt,ifl≠m,1+πllΔt+oΔt,ifl=m,where Δt>0 and oΔt/Δt⟶0 when Δt⟶0. πrk≥0 and πlm≥0 are, respectively, the transition probability rates from mode r at time t to mode k and from mode l to mode m, with ∑k=1,k≠rN1πrk=−πrr and ∑m=1,m≠lN2πlm=−πll. αt is the vector-valued initial continuous function defined on a interval −h2,0, where h2=maxh2l,l∈Λ2, and βt and γt are, respectively, the initial mode.For convenience, we denoteAirt, Adirt, Birt, Bfirt, Cirt, Dirt, and xt−dt,lt as Air, Adir, Bir, Bfir, Cir, Dir, and xt−dl, respectively. Before proceeding further, we will introduce some definitions and assumptions to be needed in the development of main results through this paper.Definition 1.
The T-S fuzzy MJSs (whenut=ft=ωt=0) are said to be stochastically stable if, for any initial xt=αt and initial mode rt=βt, lt=γt has(3)limT⟶∞E∫0Txt,αt,βt,γt2dt|xt=αt,βt,γt<∞.Definition 2.
In the Euclidean spaceℝn×Λ1×Λ2×ℝ+, we introduce a stochastic positive functional as Vxt,rt=r,lt=l,t>0, the weak infinitesimal operator of which satisfies(4)ℑVxt,r,l,t=limΔt⟶01ΔtEVxt+Δt,rt+Δt,lt+Δt−Vxt,r,l,t.Assumption 1.
(5)RankCirBfir=q.Assumption 2.
Invariant zeros ofAirBfirCir lie in open left-half plane.Remark 1.
Assumptions1 and 2 mean that the triple AirBfirCir has relative degree one and is minimum-phase. Generally speaking, Assumptions 1 and 2 hold if and only if symmetric positive definite matrices Pr,l, Q and matrices Lir,l, Hir,l exist, satisfying the following conditions:(6)Pr,lAir−Lir,lCir+Air−Lir,lCirTPr,l=−Q,BfiTrPr,l=Hir,lCir.
Therefore, Assumption1 usually holds for the cases where the number of output is greater than or equal to the number of fault inputs; Assumption 2 is satisfied for most systems [20].Remark 2.
The external disturbance inputωtis arbitrary energy bounded in the considered system (1), and then, it is not necessary to obtain its exact knowledge such as the statistics. Moreover, the time-varying delaydt,ltis mode-dependent differentiable and the delay modeltis not the same as that for jump in the system state modert. Therefore, the system state delay is the less conservative form and the application scope of the system mode is wide.Then, by using a standard fuzzy singleton inference method, i.e., a singleton fuzzifier to produce a fuzzy inference and weighted center-average defuzzifer, the T-S fuzzy dynamic MJSs can be obtained as follows:(7)x˙t=∑i=1sμitAirxt+Adirxt−dl+Birut+Bfirft+Dirωt,yt=∑i=1sμitCirxt,xt=αt,rt=βt,lt=γt,∀t∈−h2,0,in which μit=μiξt=βiξt/∑i=1sβiξt, where βiξt=∏i=1pMijξjt, and Mijξjt represents the grade of membership of ξjt in Mij. Here, we always assume that βiξt≥0, i=1,…,s, ∑i=1sβiξt>0 for any ξt. Hence, μiξt satisfy μiξt≥0,i=1,…,s,∑i=1sμiξt=1 for any ξt. In order to estimate the fault, our task is to design a suitable fault estimation observer. First, we consider the following T-S fuzzy dynamic jumping observer.Observer rulei: IF ξ1t is Mi1, and …, and ξpt is Mip, THEN(8)x^˙t=Airx^t+Adirx^t−dl+Birut+Bfirf^t+Lir,lyt−y^t+Ldir,lydt−y^dt,y^t=Cirx^t,x^t=α¯t,∀t∈−h2,0,and the global T-S fuzzy dynamic jumping observer model can be constructed as(9)x^˙t=∑i=1sμitAirx^t+Adirx^t−dl+Birut+Lir,lyt−y^t+Bfirf^t+Ldir,lydt−y^dt,y^t=∑i=1sμitCirx^t,x^t=α¯t,∀t∈−h2,0,where x^t is the observer state, y^t is the observer output, and α¯t is the initial estimation state. Here, we assume that the system matrices and mode-dependent time-varying delay of the observer have the same process as system (7). So, the initial mode value is defined as rt=βt and lt=γt, where βt and γt are, respectively, the initial mode in system (7). This assumption for fuzzy jumping systems is common in the existing literature [11, 17, 18, 39]. Denote the state and fault estimation errors as ext=xt−x^t and eft=ft−f^t, and we can obtain the following error dynamic system by combining (7) and (9):(10)e˙xt=∑i=1s∑j=1sμiξtμjξtAir−LirCjrext+Bfireft+Adir−LdirCjrext−dl+Dirωt,=A^rext+A^drext−dl+B^freft+D^rωt,eyt=∑i=1sμiξtCirext=C^rext.By comparison with the conventional adaptive fault estimation algorithm, we consider the following fast adaptive fault estimation algorithm for fuzzy Markovian jumping systems with mode-dependent time-varying state delays. In our algorithm, we addeyt in the estimation equation, that is,(11)f^t=−Γ∑i=1sμiξtHir,l∫tfteysds+eyt=−ΓH^r,l∫tfteysds+eyt,where the matrix Γ is the learning rate and tf denotes the instant when fault occurs. When ωt∈L20,∞ and ft≠0, in order to minimize the external disturbance effect on fault estimation, for T>0, we introduce the following robust H∞ performance index:(12)JT=E∫0TzTtzt−γ2ωTtωtdt,where γ>0 is a given scalar reflecting disturbance attenuation level and the signal zt=L′ext is a weighted estimation error. Then, the objective of robust H∞ fault estimation is to design observer (9) such that the error system (10) has a H∞ performance γ for all admissible disturbance, i.e., the system satisfies the following: (i) the error system (10) with ωt=0 is stochastically stable and (ii) the H∞ performance (12) is guaranteed for all nonzero ωt∈L20,∞ and the fault estimation algorithm can realize that limt⟶∞eft=0.


## 3. Main Results

In this section, the robust FE observer design method will be proposed not only to evidently improve performances of fault estimation but also to guarantee the filtering error system to be stochastically stable with a prescribedH∞ performance. The main results are presented in the following theorem.Theorem 1.
For the given scalarsγ>0 and 0<h1<h2, the error dynamic system (10) with mode-dependent time-varying delays is stochastically stable (when ωt=ft=0) and has a prescribed H∞ performance level γ (when ωt∈L20,∞) if there exist mode-dependent symmetric positive-definite matrix Pr,l>0, Q>0, Zi>0 (i = 1, 2), W>0, G>0, H^r,l, and N=N1TN2TN3T0N4T and M=M1TM2TM3T0M4T, such that the following matrix inequalities hold for all r,k∈Λ1, l,m∈Λ2:(13)H^r,lC^r=B^fTrPr,l,(14)Ξr,lh2MTh12NTh2ΓTrPr,lh12ΓTrPr,l∗−h2Z1000∗∗−h12Z200∗∗∗−h2Pr,lZ1−1Pr,l0∗∗∗∗−h12Pr,lZ2−1Pr,l<0,where h1=minh1l,l∈Λ2,h2=maxh2l,l∈Λ2,(15)Ξr,l=Ξ11Ξ12Ξ13Ξ14Ξ15∗Ξ22Ξ23Ξ24Ξ25∗∗Ξ330−N4T∗∗∗Ξ44Ξ45∗∗∗∗−γ2I,ΓTr=A^TrA^dTr0B^fTrD^Tr,Ξ11=Pr,lA^r+A^TrPr,l+∑k=1N1πrkPk,l+∑m=1N2πlmPr,m+1+πh12W+Q+L′TL′+M1+M1T,π=maxπrr,r∈Λ1,Ξ12=Pr,lA^dr−M1+M2T+N1,Ξ13=M3T−N1,Ξ14=−A^TrPr,lB^fr,Ξ15=Pr,lD^r+M4T,Ξ22=−1−h3lW−M2−M2T+N2+N2T,Ξ23=−M3T+N3T−N2,Ξ24=−A^drPr,lB^fr,Ξ25=−M4T+N4T,Ξ33=−Q−N3−N3T,Ξ44=G−2B^frPr,lB^fr,Ξ45=−B^fTrPr,lD^r.Proof. The process of proof can be classified into three parts.Part 1. Construct the Lyapunov–Krasovskii function.Part 2. Make the error dynamic (10) stochastically stable when ωt=0 and ft=0.Part 3. Find the observer gains Lir,l and Ldir,l to make that the H∞ norm of dynamic (10) is less than the prescribed value γ when the exogenous disturbance signal ωt≠0. At the same time, the fault is estimated by algorithm (11).Part 1. For the error dynamic (10), we define the stochastic Lyapunov–Krasovskii functional as Vext,r,l,t=V1ext,r,l,t+V2ext,r,l,t+V3ext,r,l,t+V4ext,r,l,t, where(16)V1ext,r,l,t=exTtPr,lext,V2ext,r,l,t=∫t−h2texTsQexsds+∫t−dt,lttexTsWexsds,V3ext,r,l,t=π∫−h2−h1∫t+θtexTsWexsdsdθ+∫−h20∫t+θte˙xTsZ1e˙xsdsdθ+∫−h2−h1∫t+θte˙xTsZ2e˙xsdsdθ,V4ext,r,l,t=efTtΓ−1eft.By recalling the weak infinitesimal operatorℑ in Definition 2, the time derivative of Vext,r,l,t along the trajectories of the overall global T-S fuzzy error dynamic systems (10) is given as(17)ℑV1ext,r,l,t=limΔt⟶01ΔtEexTt+ΔtPrt+Δt,lt+Δtext+Δt−exTtPrt,ltext=limΔt⟶01ΔtexTt+Δt1+πrrΔt+oΔt1+πllΔt+oΔtPrt,ltext+Δt−exTt+Δt1+πrrΔt+oΔt∑m=1N21+πlmΔt+oΔtPrt,mtext+Δt+exTt+Δt1+πllΔt+oΔt∑k=1N11+πrkΔt+oΔtPkt,ltext+Δt+exTt+Δt∑m=1N21+πlmΔt+oΔt∑k=1N11+πrkΔt+oΔt×Pkt,mtext+Δt−exTtPkt,ltext=2exTtPrt,ltA^rext+A^drext−dl+B^freft+D^rωt+exTt∑m=1N2πlmPrt,mt+∑k=1N1πrkPkt,ltext,ℑV2ext,r,l,t=exTtQext−exTt−h2Qext−h2+limΔt⟶01ΔtE∫t+Δt−dt+Δt,lt+Δtt+ΔtexTsWexsds−∫t−dt,lttexTsWexsds,=exTtQext−exTt−h2Qext−h2+∑m=1N2πlm∫t−dmtexTsWexsds+limΔt⟶01Δt∫t+Δt−dt,ltt−dt,ltexTsWexsds+∫tt+ΔtexTsWexsds=exTtQ+Wext−exTt−h2Qext−h2−1−d˙lexTt−dlWext−dl+∑m=1N2πlm∫t−dmtexTsWexsds.Noticing thatπlm≥0 for l≠m and πll≤0, we can get that(18)∑m=1N2πlm∫t−dmtexTsWexsds=∑l≠mπlm∫t−dmtexTsWexsds+πll∫t−dmtexTsWexsds≤∑l≠mπlm∫t−h2texTsWexsds+πll∫t−h1texTsWexsds=−πll∫t−h2texTsWexsds+πll∫t−h1texTsWexsds≤π∫t−h2t−h1exTsWexsds.Then, we denoteh12=h2−h1, and the time derivative of V3ext,r,l,t is as follows:(19)ℑV3ext,r,l,t=πh12exTtWext−π∫t−h2t−h1exTsWexsds+e˙xTth2Z1+h12Z2e˙xt−∫t−h2te˙xTsZ1e˙xsds−∫t−h2t−h1e˙xTsZ2e˙xsds≤πh12exTtWext−π∫t−h2t−h1exTsWexsds+e˙xTth2Z1+h12Z2e˙xt−∫t−dlte˙xTsZ1e˙xsds−∫t−h2t−dle˙xTsZ2e˙xsds.Based on the assumption that the derivative offt with respect to time is norm-bounded f˙t≤f1, it is easy to show that, for a symmetric positive definite matrix G, it can obtain that(20)2efTtΓ−1f˙t≤efTtGeft+f˙TtΓ−1G−1Γ−1f˙t≤efTtGeft+δ,where δ=f12·λmaxΓ−1G−1Γ−1. So, it can get the derivative of V4ext,r,l,t as(21)ℑV4ext,r,l,t=−2efTtHr,lC^re˙xt+ext+2efTtΓ−1f˙t=−2efTtB^fTrPr,lA^rext−2efTtB^fTrPr,lA^drext−dl−2efTtB^fTrPr,lB^freft−2efTtB^fTrPr,lD^rωt−2efTtB^fTrPr,lext+efTtGeft+δ.Part 2. Taking into account ωt=0 and ft=0, the weak infinitesimal operator ℑ of Vext,r,l,t can be obtained:(22)ℑVext,r,l,t=2exTtPrt,lte˙xt+exTt∑m=1N2πlmPrt,mt+∑k=1N1πrkPkt,ltext+exTtQ+Wext−exTt−h2Qext−h2−1−d˙lexTt−dlWext−dl+∑m=1N2πlm∫t−dmtexTsWexsds+πh12exTtWext−π∫t−h2t−h1exTsWexsds+h2e˙xTtZ1e˙xt−∫t−h2te˙xTsZ1e˙xsds+h12e˙xTtZ2e˙xt−∫t−h2t−h1e˙xTsZ2e˙xsds.From the Leibniz–Newton formula, the following equations are true for any matricesM′ and N′ with appropriate dimensions:(23)2ηTtM′ext−ext−dl−∫t−dlte˙xsds=0,(24)2ηTtN′ext−dl−ext−h2−∫t−h2t−dle˙xsds=0,where ηTt=exTtexTt−dlexTt−h2. By adding (23) and (24) to (22), it can obtain that(25)ℑVext,r,l,t≤ηTtΠ¯r,l+h2M′Z1−1M′T+h12N′Z2−1N′T+h2Γ2TZ1Γ2+h12Γ2TZ2Γ2ηt+∫t−dltηTtM′+e˙xTsZ1Z1−1M′Tηt+Z1Te˙xsds+∫t−h2t−dlηTtN′+e˙xTsZ2Z2−1N′Tζt+Z2Te˙xsds,where(26)Π¯r,l=Ξ11Ξ12Ξ13∗Ξ22Ξ23∗∗Ξ33,Γ2Tr=A^TrA^dTr0,M′N′=M1N1M2N2M3N3.BecauseZi=ZiT>0i=1,2, then(27)ηTtM′+e˙xTsZ1Z1−1M′Tηt+Z1Te˙xs≥0and ηTtN′+e˙xTsZ2Z2−1N′Tηt+Z2Te˙xs≥0 hold. Therefore, if there exists a feasible solution to Π¯r,l+h2M′Z1−1M′T+h12N′Z2−1N′T+h2Γ2TZ1Γ2+h12Γ2TZ2Γ2<0, we can conclude that ℑVext,r,l,t<0 when ωt=0 and ft=0. It is obviously that ℑVext,r,l,t<0 can be guaranteed by matrix inequality (14). And, if condition (14) holds, there will exist matrix Ξr,l>0, such that ℑVext,r,l,t=−ηTtΞr,lηt<0. In addition, since ℑVext,rt,lt,t<0, we can get Vext,rt,lt,t<Vex0,r0,l0,t|t=0. Then, by defining M1=inf−τ2<δ1<tEηδ12, σΞ=minr∈Λ1,l∈Λ2σminΞr,l, and M2=sup−τ2<δ2<tEexδ22, the following relation holds:(28)ℑVext,rt,lt,tVext,rt,lt,t<−ηTtΞr,lηtVex0,r0,l0,t|t=0≤−M1σΞM2σΣ=−σ,where σΣ is the sum of a set of positive numbers, σ>0. So, it follows that ℑVext,rt,lt,t<−ϱVext,rt,lt,t. That is, EVext,rt,lt,t<exp−σtEVex0,r0,l0,t=0. Then, by letting ρ=σΣM2, for a given small positive scalar ε>0, we can get εEexTtext≤EVext,rt,lt,t≤ρexp−σt. Taking limit as T⟶∞, it follows that(29)limT⟶∞E∫0TexTtextdt|ex0=α0,r0,l0≤limt⟶∞ρε1−exp−σt=ρε<∞.By recalling Definition1, we know that the overall global T-S fuzzy error dynamic system (10) is stochastically stable.Part 3. On the contrary, when ωt∈L20,∞ and ft≠0, to attain the robustness of the proposed observer to the bound disturbances, we define(30)V0=zTtzt−γ2ωTtωt+V˙ext,r,l,t.From the Leibniz–Newton formula, the following equations are true for any matrices with appropriate dimensions:(31)2ζTtMext−ext−dl−∫t−dlte˙xsds=0,(32)2ζTtNext−dl−ext−h2−∫t−h2t−dle˙xsds=0,where ζTt=exTtexTt−dlexTt−h2efTtωTt, M=M1TM2TM3T0M4TT, and N=N1TN2TN3T0N4TT. By adding (31) and (32) to (30), we can obtain that(33)V0≤exTtL′TL′ext−γ2ωTtωt+exTtPrt,ltA^r+A^TrPrt,lt+∑k=1N1πrkPkt,lt+∑m=1N2πlmPrt,mtext+exTtQ+1+πh12Wext+2exTtPr,lA^drext−dl+2exTtPr,lB^freft+2exTtPr,lD^rωt−exTt−h2Qext−h2−1−d˙lexTt−dlWext−dl−2efTtB^fTrPr,lA^rext−2efTtB^fTrPr,lA^drext−dl−2efTtB^fTrPr,lB^freft−2efTtB^fTrPr,lD^rωt−2efTtB^fTrPr,lext+efTtGeft+δ+2ζTtMext−ext−dl−∫t−dlte˙xsds+2ζTtNext−dl−ext−h2−∫t−h2t−dle˙xsds+e˙xTth2Z1+h12Z2e˙xt−∫t−dlte˙xTsZ1e˙xsds−∫t−h2t−dle˙xTsZ2e˙xsds≤ζTtΠr,l+h2MZ1−1MT+h12NZ2−1NT+h2Γ1TZ1Γ1+h12Γ1TZ2Γ1ζt+δ+∫t−dltζTtM+e˙xTsZ1Z1−1MTζt+Z1Te˙xsds+∫t−h2t−dlζTtN+e˙xTsZ2Z2−1NTζt+Z2Te˙xsds.Hence, whenωt≠0 and ft≠0, if there exists a feasible solution to ζTtΠr,l+h2MZ1−1MT+h12N2−1NT+h2A1TZ1A1+h12A1TZ2A1ζt+δ<0, it can conclude that(34)V0=zTtzt−γ2ωTtωt+V˙<0.We assume thatε is the minimum eigenvalue of −Πr,l+h2MZ1−1MT+h12N2−1NT+h2A1TZ1A1+h12A1TZ2A1. Therefore, when εζt2>δ, equation (34) holds. Πr,l+h2MZ1−1MT+h12N2−1NT+h2A1TZ1A1+h12A1TZ2A1<0 is equivalent to (14) by Schur complements. This implies the H∞ performance index J<0 is satisfied under zero initial condition. The proof is completed.Remark 3.
In order to further reduce the conservativeness of FE criteria for mode-dependent time-varying delay MJSs, two different Markov stochastic processes are considered: one is the state modert and the other is time delay mode lt. Moreover, when giving FE analysis results for MJSs, the term ∫t−h2texTsQexsds was ignored in [17], or some useful integral term ∫−h2−h1∫t+θte˙xTsZ2e˙xsdsdθ was lost, e.g., [18]. In this paper, by constructing the new Lyapunov–Krasovskii functional candidate, which make full use of the information of both lower and upper bounds of delay dt,lt, the less conservative result than the existing ones in [17, 18] is obtained.Remark 4.
The fast adaptive fault estimation algorithm to estimate the fault has been developed in [17, 20, 21], but it is not extended to the fuzzy MJSs system with mode-dependent interval time-varying delay, unknown actuator fault, and exogenous disturbance signal simultaneously. If there is only one Markovian process rt and the type of time delay is constant in system, then system (7) in this paper reduces to the existing one in [17]. And, if the fuzzy rule number s=1, (7) can be transformed to the one in [18].Remark 5.
The effect of fuzzy MJSs uncertainties on the estimation errors is decided by the value ofγ. The accuracy of the fault estimation increases with smaller value of γ. The minimization of γ can be found by solving the LMI optimization problem, which has been applied broadly. However, since the term h2PZ1−1P and h12PZ2−1P exist simultaneously, Theorem 1 cannot be solved directly. Next, we will choose an inequality transformation method to give the strict LMIs’ conditions.Theorem 2.
For the given scalarsγ>0 and 0<h1<h2, the error dynamic system (10) with mode-dependent time-varying delays is stochastically stable (when ωt=ft=0) and has a prescribed H∞ performance level γ (when ωt∈L20,∞) if there exist a set of mode-dependent symmetric positive-definite matrix Pr,l>0, Q>0, W>0, Z1>0, Z2>0, G>0, Hir,l, and N=N1TN2TN3T0N4T and M=M1TM2TM3T0M4T, such that the following matrix inequalities hold for all r,k∈Λ1, l,m∈Λ2. Minimize ρ>0 which subjects to(35)ρIBfqTrPr,l−Hir,lCir∗ρI>0,(36)Φiiir,l<0,1≤i≤s,(37)Φijqr,l+Φjqir,l+Φqijr,l<0,1≤i≤j<q≤s,(38)Φqjir,l+Φjiqr,l+Φiqjr,l<0,1≤i<j≤q≤s,where(39)Φijqr,l=ϕ11ϕ12Ξ13ϕ14ϕ15h2M1h12N1ϕ18ϕ19∗Ξ22Ξ23ϕ24Ξ25h2M2h12N2ϕ28ϕ29∗∗Ξ330−N4Th2M3h12N300∗∗∗ϕ44ϕ4500ϕ48ϕ49∗∗∗∗−γ2Ih2M4h12N4ϕ58ϕ59∗∗∗∗∗−h2Z1000∗∗∗∗∗∗−h12Z200∗∗∗∗∗∗∗ϕ880∗∗∗∗∗∗∗∗ϕ99,ϕ11=Pr,lAir+AiTrPr,l−Yir,lCjr−CjTrYiTr,l+∑k=1N1ΠrkPk,l+∑m=1N2ΠlmPr,m+Q+1+πh12W+L′TL′+M1+M1T,ϕ12=Pr,lAdir−Ydir,lCjr−M1+M2T+N1,ϕ14=CjTrYiTr,lBfqr−AiTrPr,lBfqr,ϕ15=Pr,lDir+M4T,ϕ24=CjTrYdiTr,lBfqr−AdiTrPr,lBfqr,ϕ44=G−2BfqTrPr,lBfqr,ϕ45=−BfqTrPr,lDir,ϕ18=h2AiTrPr,l−CjTrYiTr,l,ϕ19=h12AiTrPr,l−CjTrYiTr,l,ϕ28=h2AdiTrPr,l−CjTrYdiTr,l,ϕ29=h12AdiTrPr,l−CjTrYdiTr,l,ϕ48=h2BfqTrPr,l,ϕ49=h12BfqTrPr,l,ϕ58=DirPr,l,ϕ59=DirPr,l,ϕ88=h2−2σPr,l+σ2Z1,ϕ99=h12−2σPr,l+σ2Z2.
Moreover, the fast adaptive fault estimation algorithm (11) can realize ext and eft uniformly ultimate bounded and the jumping fault estimation observer gain can be obtained by Lir,l=P−1r,lYir,l and Ldir,l=P−1r,lYdir,l.Proof. For any scalar σ, it follows from the fact σZi−Pr,lZi−1σZi−Pr,l≥0 that(40)−Pr,lZi−1Pr,l≤−2σPr,l+σ2Zi,i=1,2.Then, considering the error dynamic system (10), we can conclude that condition (14) holds if the following inequality holds:(41)∑i=1sμit∑j=1sμjt∑q=1sμqtΦijqr,l<0,where(42)Φijqr,l=Ξr,lh2MTh12NTh2ΓTrPr,lh12ΓTrPr,l∗−h2Z1000∗∗−h12Z200∗∗∗ϕ880∗∗∗∗ϕ99,Ξr,l=ϕ11ϕ12Ξ13ϕ14ϕ15∗Ξ22Ξ23ϕ24Ξ25∗∗Ξ330−N4T∗∗∗ϕ44ϕ45∗∗∗∗−γ2I,Γ^Tr=AiTrPr,l−CjTrPr,lLir,lTAdiTrPr,l−CjTrPr,lLdir,lT0BfqTrPr,lDirPr,l,ϕ11=Pr,lAir−Pr,lLir,lCjr+AiTrPr,l−CjTrLiTr,lPr,l+∑k=1N1ΠrkPk,l+∑m=1N2ΠlmPr,m+Q+1+πh12W+L′TL′+M1+M1T,ϕ12=Pr,lAdir−Pr,lLdir,lCjr−M1+M2T+N1,ϕ14=CjTrLiTr,lPr,lBfqr−AiTrPr,lBfqr,ϕ15=Pr,lDir+M4T,ϕ24=CjTrLdiTr,lPr,lBfqr−AdiTrPr,lBfqr,ϕ44=G−2BfqTrPr,lBfqr,ϕ45=−BfqTrPr,lDir.It can be easily shown that inequality (41) is equivalent to the following condition:(43)∑i=1sμi3Φiiir,l+∑i=1sμi∑i≤jsμj∑j<qsμqΦijqr,l+Φjqir,l+Φqijr,l+∑i=1sμi∑i<jsμj∑j≤qsμqΦqjir,l+Φjiqr,l+Φiqjr,l<0.By lettingYir,l=Pr,lLir,l and Ydir,l=Pr,lLdir,l, it leads to the LMIs (36)–(38). On the contrary, it is hard to solve (36 – 38) with (13) simultaneously, and we can transform it into the optimization problem (35). This completes the proof.Remark 6.
Since condition (13) in Theorem 1 is no longer LMI, it cannot be solved directly using the LMI toolbox of Matlab because of the term PZi−1P. In [20], by defining new variables Si such that PZi−1P≥Si, an alternative way is proposed to transform the conditions into LMIs based on nonlinear minimization problem using the cone complementary linearization algorithm. However, more new variables H, U, and Kii=1,2 are introduced in the result, which must meet UiHHKi≥0, H=P−1, Ui=Si−1, and Ki=Zi−1. In this paper, by using the inequality transformation method, we can easily obtain the LMIs’ conditions without introducing any new variables. This can reduce the computational complexity.Remark 7.
It is necessary to point out that there will bring some difficulties when usingTheorem 1 to study the fault estimation observer design problems for fuzzy MJSs due to the existence of equality constraints (13). In order to make BfqTrPr,l approximate to Hir,lCir with a satisfactory precision, a sufficiently small positive scalar ρ should be selected in advance to meet condition (35). We can get the solutions of Theorem 2 by solving an optimization problem with (35), and it is straightforward to check the feasibility of LMIs by using LMIs’ toolbox.Remark 8.
You et al. have proposed, in [20], an adaptive observer to estimate the actuator fault-affecting linear systems. The similar work has been extended by He in [17] for T-S fuzzy Markovian jumping systems. The restriction of result in [17] lies in three points. The first one is that the result is about the mode independent state delay. The second one is that the delay must be constant, not considering the interval time delay. The third one is that the external disturbance effect is nonincluded. The main advantage of the proposed design in Theorem 2 is the reduction of conservatism by presenting a mode-dependent time delay result. Also, a time-varying delay has been considered. Furthermore, the design procedure gives the robust fault estimation observer design method.Remark 9.
It should be mentioned that the results proposed in Theorem2 are built under the conditions that T-S fuzzy systems are about type-1 fuzzy sets and FE observer share the same membership function of the plants. However, in practice, when uncertainties exist in nonlinear plant or the membership functions, the proposed FE method will fail. To overcome this problem, in literature [40], a state feedback nonfragile controller is designed for interval-valued fuzzy systems against nonlinear actuator fault. The obtained results in [40] can better solve the nonfragile control problem for interval-valued fuzzy systems under the condition that different membership functions exist. However, in this situation, how to apply the proposed FE approach to interval-valued T-S fuzzy systems becomes meaningful but also complicated, which will be researched in the future.


## 4. Numerical Examples

In this section, three examples are given to demonstrate the effectiveness of the proposed methods in this paper.Example 1.
We present the following T-S fuzzy MJSs with two jumping operation modes:(44)A11=−0.5−0.3−0.1−1,Aτ11=0.05−0.10−0.05,C1r=C2r=0.60,A12=−0.5−0.4−0.1−1.06,Aτ12=0.07−0.10−0.05,D1r=D2r=0.50,A21=−0.5−0.3−0.1−1,Aτ21=−0.050.100.05,B1r=B2r=1.00.5,A22=−0.5−0.4−0.1−1.06,Aτ22=−0.070.100.05,Bf1r=Bf2r=0.10.1.
In this paper, we assume thatx1t∈−0.50.5 and represent the membership functions of h1x1t and h2x1t as h1x1t=0.51−2x1t and h2x1t=0.51+2x1t. Meanwhile, we assume state time delay have two changed modes when fuzzy plant rule i=1=2, respectively, d1t,1=d2t,1=0.2sint+0.4 and d1t,2=d2t,2=0.5t+0.3, so we can get the value h11=0.2, h21=0.6, h12=0.3, and h22=0.8 and h1=minh1l,l∈Λ2=0.2 and h2=maxh2l,l∈Λ2=0.8 with Λ2=1,2, d˙1t,l=0.2cost≤h31=0.2, and d˙2t,l=h32=0.5. The transition rate matrix that relates the two operation modes are, respectively, given as Π1=−0.50.50.4−0.4 and Π2=−0.50.50.4−0.4.
Then, for the fixed scalarσ=1 and set γ=1.5, ρ=0.001 applying the solver of LMI toolbox in Matlab, and we can obtain the corresponding feasible solutions based on Theorem 2 as follows:(45)P1,1=3.8752−0.3151−0.31513.8587,P1,2=3.8320−0.3153−0.31533.8164,P2,1=3.6467−0.3616−0.36163.9573,P2,2=3.6508−0.3613−0.36133.9611,Y11,1=2.2784−0.6306,Yd11,1=0.6243−0.0011,Y11,2=2.2381−0.6231,Yd11,2=0.6232−0.0026,Y12,1=2.0983−0.6687,Yd12,1=0.6927−0.0417,Y12,2=2.1030−0.6690,Yd12,2=0.6928−0.0416,Y21,1=2.1156−0.6881,Yd21,1=0.10130.1017,Y21,2=2.0816−0.6770,Yd21,2=0.10560.0998,Y22,1=1.9343−0.7369,Yd22,1=−0.02670.1021,Y22,2=1.9384−0.7376,Yd22,2=−0.03130.1022,(46)L11,1=0.5785−0.1162,Ld11,1=0.16220.0130,L11,2=0.5746−0.1153,Ld11,2=0.16370.0129,L12,1=0.5638−0.1175,Ld12,1=0.19060.0069,L12,2=0.5644−0.1174,Ld12,2=0.19050.0069,L21,1=0.5350−0.1346,Ld21,1=0.02850.0287,L21,2=0.5322−0.1334,Ld21,2=0.02990.0286,L22,1=0.5166−0.1390,Ld22,1=−0.00480.0254,L22,2=0.5172−0.1390,Ld22,2=−0.00610.0252.
G=0.0126, and Hir,l=0.5687i=1,2. Then, by the results of Theorem 2, the observer gains Lir,l and Ldir,l are obtained as follows:
For the simulation, we assume that the abrupt fault signal is created as(47)f1t=0,0≤t<5,5,5≤t≤20,f2t=00≤t<5,2sint5≤t≤20.
By taking the learning rateΓ=100 and assuming ωt is band-limited white noise with sampling time 0.01 s and noise power is 0.1 and 5, respectively. Such an assumption remains the same for the next simulation examples. Then, using the fault estimation approach in Theorem 2, simulation results shown in Figures 1 and 2 obviously illustrate that the proposed FE method have a good performance to estimate the system fault. Meanwhile, it can be seen from Figures 3 and 4 that the error dynamic system has a small fluctuation when ωt≠0 and it is also stochastic stable when the disturbance input ωt does not exists.Figure 1
Response curves of constant faultf1t and its estimation.Figure 2
Response curves of time-varying faultf2t and its estimation.Figure 3
Response curves of error dynamic statese1t and e2t when ωt≠0.Figure 4
Response curves of error dynamic statese1t and e2t when ωt=0.Example 2.
For comparison purpose, next, we consider a computer simulated truck-trailer system borrowed from [22, 23]:(48)x˙1t=−vt¯Lt0x1t+vt¯lt0ut,x˙2t=vt¯Lt0x1t,x˙3t=vt¯t0sinx2t+vt¯2Lx1t.
Then, systemx1t is perturbed by time delay, and the delayed model is given as(49)x˙1t=−avt¯Lt0x1t−1−avt¯Lt0x1t−τt+vt¯lt0ut,x˙2t=avt¯Lt0x1t+1−avt¯Lt0x1t−τt,x˙3t=vt¯t0sinx2t+avt¯2Lx1t+1−avt¯2Lx1t−τt.
The constanta is the retarded coefficient, which satisfies the conditions: a∈0,1. The limits 1 and 0 correspond to no delay term and to a completed delay term, respectively. In this example, the model parameters is given as a=0.7, l=2.8, L=5.5, v=−1.0, t¯=2.0, and t0=0.5.
Then, we use the following fuzzy models to design the fuzzy fault observer:Rule 1: IFξ1t=x2t+avt¯/2Lx1t+1−avt¯/2Lx1t−τ is about 0, THEN(50)x˙t=A1xt+Ad1xt−dt+B1ut,yt=C1xt.Rule 2: IFξ1t=x2t+avt¯/2Lx1t+1−avt¯/2Lx1t−τ is about π or −π, THEN(51)x˙t=A2xt+Ad2xt−dt+B2ut,yt=C2xt.
Thus, the delay model with faultft and disturbance ωt is given by T-S fuzzy systems as follows:(52)x˙t=∑i=12μiξtAixt+Adixt−dt+Biut+Bfift+Diωt,yt=∑i=12μiξtCixt,where xt=x1tx2tx3tT and(53)A1=−avt¯Lt000avt¯Lt000av2t¯22Lt0vt¯t00,Ad1=−1−avt¯Lt0001−avt¯Lt0001−av2t¯22Lt000,B1=vt¯lt000,A2=−avt¯Lt000avt¯Lt000adv2t¯22Lt0dvt¯t00,Ad2=−1−avt¯Lt0001−avt¯Lt0001−adv2t¯22Lt000,B2=vt¯lt000.
Here, because the actuator faults usually occur in the input channel, we assume thatBi=Bfii=1,2 and C1=C2=−20.05−0.15 and the disturbance distribution matrices are D1=D2=0.0500T. For simulation purpose, we choose membership functions for Rules 1 and 2 are μ1ξt=1/1+expx1t+0.5 and μ2ξt=1−μ1ξt with initial condition 0.5π0.75π−5T and set d=10∗t0/π.
It should be noted that the abovementioned time-delay fuzzy system is a special case that we considered in this paper, which do not have mode change both system matrices and state delay under different fuzzy rules. So, we assume the time-varying delay acting on the system state is given bydt=0.2sint+0.3h1=0.1,h2=0.5,h12=0.4,h3=0.2 which are the same as [22]; then, by computing matrix inequalities (35)–(38) in Theorem 2 based on Matlab LMIs toolbox, one obtains a feasible solution when γ=3:(54)Y1=0.0473−0.0850−0.003,Yd1=0.0058−0.0369−0.0001,Y2=0.0522−0.0933−0.0003,Yd2=0.0063−0.0405−0.0001,L1=3.2976−0.9125−1.8335,Ld1=0.4022−0.39990.3674,L2=3.6335−1.0041−1.4626,Ld2=0.4364−0.44010.6833.
Then, a time-varying fault is simulated as(55)ft=0,0≤t<5,sin3t−9+sin4t−12,5≤t≤20.
Figure5 illustrates fault estimation simulation results. It is obvious that despite the existing initial error, the fault is estimated with satisfactory accuracy and rapidity with time. It follows from Figure 6 that the error states ex1t, ex2t, and ex3t are also stable when ωt not exists. On the contrary, the estimation error shown in Figure 7 fluctuates in a limited range when ωt≠0.Figure 5
Response curves of time-varying faultft and its estimation.Figure 6
Response curves of error dynamic statesex1 (t), ex2 (t), and ex3 (t) when ω(t) = 0.Figure 7
Response curves of error dynamic statesex1 (t), ex2 (t), and ex3 (t) when ω(t) ≠ 0.Remark 10.
In the existing literature [22], by means of the adaptive observer, the robust fault estimation for time delay T-S fuzzy systems is solved. However, it is worth mentioning that the works in [22] have considered no mode change case that the system matrices and time delay have fixed value under different fuzzy rules. However, this is not the situation always because, in practice, the fixed value cannot describe the system well, and the descriptions with mode changes are more general. Consider this characteristic of system, and the FE strategy in [22] is not valid. From Figure 5, we can see that, on one hand, the proposed method can accurately realize fault estimation under disturbance input despite there being an error in the initial stage of fault estimation. It means that whether the system and delay have mode change or not, the fault estimator always works to give the fault estimation information.Remark 11.
On the contrary, free-weighting matrix method is a common method when dealing with integral term of the derivative of Lyapunov functional candidate; however, what followed is the increasing of computational complexity. In [22], there are 18 unknown matrices in Theorem 3.1 to be solved, and they are P, Q1, Q2, G, R, Z, L, R¯, Mi, and Ni, i=1,2,3,4,5, where Mi and Ni are free-weighting matrix. The higher the dimension of the system, the more the variables of LMIs. If mode switching is not considered, there are 15 unknown matrices in Theorem 2 including free-weighting matrices Mi and Ni, i=1,2,3,4. Thus, by combining with the model and calculation of characteristics of the proposed approach, we can see that the multimode change FE method is more simpler and can be used in more field of application.Remark 12.
Generally speaking, a large number of fuzzy rules are usually needed for accurate approximation, but which may lead inefficiency or difficulty in system computation. When the number of fuzzy rules is increased, the number of LMIs to be solved increases at the same time. If fuzzy rulesr=2, there are eight LMIs in Theorem 3.1 of literature 22. However, only five LMIs are required to obtain the gain matrices by using the proposed approach in this paper. When r=3 and r=4, there are 15 and 24 LMIs to be solved in Theorem 3.1. In Theorem 2, we just need to calculate 11 and 19 LMIs, respectively. From the comparison results in Table 1, we can see that fewer decision variables and LMIs to be solved will reduce the computational complexity.Table 1
Comparison of the number of variables and LMIs to be solved.
The number of variables and LMIs to be solvedFuzzy rulesr=2r=3r=4Decision variables408152418Our results5111915Reduction ratio37.5%26.7%20.8%16.7%Example 3.
We consider the following example of a single-link robot arm model presented in [11]:(56)θ..t=−MglJsinθt−DJθ.t+1Jut+LJωt,where θt is the angle position of the arm, ut is the control input, ωt is the unknown disturbance, M is the mass of the payload, g is the acceleration of gravity, L is the length of the arm, J is the moment of inertia, and D is the coefficient of viscous friction. The values of parameters g, D, and L are given by g=9.8, D=0.2, and L=0.5, respectively. It is also supposed that the system parameters M and J have two different modes as [17], when system mode r=1, set M=1 and J=1, and when system mode r=2, we set M=5 and J=5. And, the transition rate matrices that relate the two operation modes are given as(57)Π1=−0.50.50.4−0.4,Π2=−0.80.80.6−0.6.
Here, we let the system statext=x1t,x2t=θt,θ.t, where x2t denotes the time rate change of the angle position of arm. When θt changes, x2t=θ˙t will change accordingly. However, in this process, there will be undesired phenomena, such as time delay. In [41], it is assumed that x2t is perturbed by one fixed type time-varying delay to study the H∞ control problem. However, there will be a certain degree inaccuracy by means of this kind of time delays. In order to better describe the randomness of time delay, it is necessary to consider the mode-dependent time delays when single-link robot arm system modes switch randomly from one mode to another. Under the condition −179.4270<θt<179.4270, the nonlinear term sinθt can be represented as sinx1t=h1x1t+h2x1tλx1t, with λ=1/100π, where h1x1t, h2x1t∈0,1, and h1x1t+h2x1t=1. Then, using the membership function h1x1t and h2x1t, we can get the time-delay nonlinear MJSs with actuator fault by the following T-S models for the single-link robot arm with the selected parameter λ:Plant rule 1: IFx1t is about 0 rad, THEN(58)x˙t=A1rxt+Ad1rxt−dt,l+B1rut+Bf1rft+D1rωt.Plant rule 2: IFx1t is about π rad or −π rad, THEN(59)x˙t=A2rxt+Ad2rxt−dt,l+B2rut+Bf2rft+D2rωt,where the system matrices of two different modes are as follows:(60)A11=01−gL−D,Ad11=0.05−0.10−0.05,B11=B21=10.5,A12=01−gL−0.8D,Ad12=0.07−0.10−0.05,Bf11=Bf21=00.5,A21=01−λgL−D,Ad21=0.05−0.10−0.05,B12=B22=10.5,A22=01−gL−0.8D,Ad22=0.07−0.10−0.05,Bf12=Bf22=00.5,C1r=C2r=0.60, and D1r=D2r=0.10T. With the initial value for G=0.5, σ=10, and ρ=0.001 and the bounds of time delay remaining the same as example 1, when the attenuation value γ=1.5, one can obtain the feasible solutions by computing the matrix inequalities (35)–(38) in Theorem 2.
Wheni=1,(61)Y11,1=31.7772−8.6152,Yd11,1=0.8943−0.0723,Y11,2=31.4923−8.6142,Yd11,2=0.8873−0.0723,Y12,1=31.0695−8.5915,Yd12,1=0.9397−0.0648,Y12,2=31.0937−8.5915,Yd12,2=0.9413−0.0649.
Wheni=2,(62)Y21,1=268.939324.5274,Yd21,1=−2.8653−0.2155,Y21,2=268.668224.4924,Yd21,2=−2.8622−0.2152,Y22,1=282.232525.6010,Yd22,1=−4.8811−0.3422,Y22,2=269.198022.0768,Yd22,2=−3.3712−0.2226.
Then, by Theorem2, the observer gains Lir,l and Ldir,l are obtained as follows.
Wheni=1,(63)L11,1=4.72734.4522,Ld11,1=0.0996−0.0325,L11,2=4.72704.4491,Ld11,2=0.0997−0.0325,L12,1=4.71304.6485,Ld12,1=0.1384−0.0308,L12,2=4.70564.3736,Ld12,2=0.1425−0.0278.
Wheni=2,(64)L21,1=4.76994.4474,Ld21,1=−0.0524−0.0222,L21,2=4.76924.4447,Ld21,2=−0.0523−0.0222,L22,1=4.75154.6431,Ld22,1=−0.0856−0.0238,L22,2=4.73404.3726,Ld22,2=−0.0616−0.0155.
For the simulation, we assume that constant faultf3t and time-varying fault f4t are, respectively, created as(65)f3t=0,0≤t<5,51−e−t−5,5≤t≤20,f4t=0,0≤t<5,15sin0.5t+sint,5≤t≤10.
It can be seen from Figures8 and 9 that whether the faults are constant or time-varying, the estimation of system faults can be obtained accurately and timely. When the disturbance input exist ωt≠0, the error dynamic system states fluctuate in a small range which are shown in the simulation result Figure 10, and when ωt=0, the stochastic stability of the error system is guaranteed in Figure 11. The purpose of simulation is to verify the robustness of the proposed method with respect to modeling fault estimation. In order to give a comparison with the small disturbance fault estimate, relatively larger disturbance power is given in Examples 1 and 3 and the other parameters remain the same as before. As indicated by the simulation results shown in Figures 1 and 2 and Figures 8 and 9, we can see that the proposed method can realize accurate fault estimation under disturbance input. This illustrates the effectiveness of the approach proposed in this paper.Figure 8
Response curves of constant faultf3t and its estimation.Figure 9
Response curves of time-varying faultf4t and its estimation.Figure 10
Response curves of error dynamic statese1t and e2t when ωt≠0.Figure 11
Response curves of error dynamic statese1t and e2t when ωt=0.


## 5. Conclusion

In this paper, we have studied the design of adaptive fault estimation observer for a class of T-S fuzzy Markovian jump systems with mode-dependent interval time-varying delays. By selecting the appropriate Lyapunov–Krasovskii functional and applying LMIs optimization techniques, sufficient conditions are derived such that the error dynamic are stochastically stable and the designed observer ensures the rapidity and accuracy performance of fault estimation. Simulation results illustrate the effectiveness of the proposed approach. It should be noted that the approach presented in this paper requires that the FE observer have the same membership functions as the plants model and not consider the interval-valued fuzzy systems. Nevertheless, the uncertainties exist in membership functions may lead to the inefficiency of the proposed FE methods. In addition, how to solve the fault-tolerant control problem under such conditions becomes particularly important. We will consider these problems in our future work.



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