Distributed fault estimation of nonlinear dynamical systems over sensor networks: a non-fragile approach

Abstract

Purpose

The purpose of this paper is to investigate the fault estimation issue of nonlinear dynamical systems via distributed sensor networks. Furthermore, based on the communication topology of sensor networks, the nonfragile design strategy considering the gain fluctuation is also adopted for distributed fault estimators.

Design/methodology/approach

By means of intensive dynamical model transformation, sufficient conditions with disturbance attenuation performance are established to design desired fault estimator gains with the help of convex optimization.

Findings

A novel distributed fault estimation framework for a class of nonlinear dynamical systems is established over a set of distributed sensor networks, where sampled data of sensor nodes via local information exchanges can be used for more efficiency.

Originality/value

The proposed distributed fault estimator gain fluctuations are taken into account for the nonfragile strategy, such that the distributed fault estimators are more applicable for practical sensor networks implementations. In addition, an illustrative example with simulation results are provided to validate the effectiveness and applicableness of the developed distributed fault estimation technique.

Introduction

As a promising technology with rapid development of signal processing and information collection, the sensor networks have been extensively studied and been successfully used in many fields, such as power systems (Gungor et al., 2010), manufacturing systems (Ota and Wright, 2006), medical systems (Kumar and Lee, 2011) and so on (Akyildiz et al., 2002; Elhabyan et al., 2019). Generally speaking, sensor networks are consisted of a group of sensors that can work collectively with information exchanges via communication network. Under this context, sensors networks can achieve distinguishing advantages compared with traditional single sensor deployments, which include more efficiency, more robustness and more reliability (Zhu et al., 2013; Chen et al., 2014; Wang et al., 2015). Senor faults or failures can also be coped with by sensor networks. This is due to two significant aspects in sensor networks: the communication topology that can network all the sensor nodes and the sensor ability that can collect the data. In particular, among various structure framework of sensor networks, the distributed structure of sensor networks has attracted high research intentions due to its local communication ability between the neighboring nodes with direct or undirect network topologies. In comparison with some centralized designs, the distributed configuration for sensor networks can obtain more communication efficiency and robustness while using less communication resources and design complexities. However, it is noticed that the distributed sensors networks also would increase the system analysis and synthesis difficulties between the sensor nodes and their neighboring nodes (Chen et al., 2014; Shahrampour et al., 2015), since the overall dynamics of the sensor networks are becoming more complicated with interactions. Encouragingly, several remarkable schemes of distributed sensor networks have been proposed with effective academic and industrial applications (Qi et al., 2001; Leung et al., 2008; Kaplan, 2006).

On another active research front line, much effort has been devoted to the so-called fault estimation and fault detection problems to deal with the increasing requirements for system dynamical safety and reliability in many practical areas (Lan and Patton, 2016; Zhang et al., 2009; Li et al., 2017; Xu et al., 2023). More precisely, based on the dynamical system model, the signal processing, and the driven data, the certain faults can be extracted by designed fault detection approaches accordingly, such that the reliable system dynamics can be considerably guaranteed by generating the fault alarm in time (Isermann, 1997; Isermann, 1984; Chen et al., 2023; Zhu and Zheng, 2023). Indeed, by using the sensing information based on sensor nodes, the fault estimation strategy could judge the potential faults according to the prescribed fault thresholds. Considering the advances of fault estimation technologies, plentiful research methods have been proposed for fault estimation issues of dynamical and complex systems. Furthermore, as nonlinear dynamical features are ubiquitous in most practical systems, the fault estimation problems of nonlinear dynamical systems are more challenging during the fault estimator design procedures, especially for some cases with high performance demands with uncertainties or disturbances (Yang et al., 2015; Li et al., 2016; Li et al., 2015). As such, it is more reasonable and significant to investigate the fault estimation problems of nonlinear dynamical systems while considering more practical design conditions. Followed by this concept, some fault estimation approaches with a single sensor utilization would have certain limitations with sensing ability and robustness. In fact, for the sensor networks, it should be pointed out that there still remain main difficulties in designing distributed fault estimators based on communication topology. To the best of the authors’ knowledge, although some pioneers works on distributed state estimation or filtering explorations with sensor networks have been made with desired results (Muhammed and Shaikh, 2017; Dong et al., 2014; Dong et al., 2016), the distributed fault estimations of nonlinear dynamical systems over sensor networks have not been fully studied yet, which mainly motivates us for this current work.

So far, the nonfragile design strategies for dynamical systems have focused on achieving more robustness for practical applications, which could help bridge the theoretical designs to the unideal implementations with certain disturbances or interruptions (Lan and Zhou, 2013; Xia et al., 2018; Yang et al., 2019). As a result, since theoretical developed controller or estimator gain fluctuations should not be neglected, many nonfragile control, state estimation, synchronization issues of dynamical systems have been widely discussed and reported in the literature. On the other hand, for the applications of sensor networks, it is also applicable to take the nonfragile framework into account for each nodes to ensure the design effectiveness. By taking into account the gain fluctuations in design processes, the derived synthesis results for dynamical systems are more applicable. However, it is noteworthy that for the distributed fault estimation and fault detection problems, as the fact that there are more than one estimator based on the sensor networks, it is more complicated when adopting the nonfragile framework in the fault estimation problem with a fully distributed manner. Unfortunately, until now, there are few results that have been addressed on this opening research area, which is still a challenging topic.

Based on the aforementioned discussions, in this paper, the distributed fault estimation issue for the nonlinear dynamical systems over sensor networks is concerned by developing a novel nonfragile approach. Compared with most existing works, our contributions can be mainly summarized as the following threefold:

A novel distributed fault estimation framework for a class of nonlinear dynamical systems is established over a set of distributed sensor networks, where the sampled data of each sensor node via local information exchanges can be used for more efficiency.

The fault estimator gain fluctuations of the sensor networks are taken into account for the nonfragile strategy, such that the distributed fault estimators are more applicable for practical sensor networks implementations.

The computational distributed fault estimator gains can be design with feasible solutions by a convex optimization method in the form of linear matrix inequality, while the desired H∞ performance can also be achieved to deal with the external disturbance attenuation.

The rest part of this work is arranged as follows. Section 2 gives some preliminaries of sensor networks and formulates the distributed fault estimation problem with the nonfragile design. Section 3 derives the main theoretical results with analysis and synthesis details. Section 4 illustrates the developed distributed estimator design method with simulation example validations and Section 5 draws the overall conclusions with future research perspective.

Standard notations are used in this paper:

ℝn denotes the n dimensional Euclidean space. I is the unit matrix with proper dimension. A−B>0 implies that A – B is positive definite. diag{⋯} stands for the block-diagonal matrix. ⊗ presents the Kronecker product. * means the ellipsis terms in symmetry matrices.

Preliminaries and problem formulation

Sensor networks

In this paper, as depicted in a directed graph, a set of N sensor nodes are grouped as a sensor network that are connected by a communication network, such that local information exchanges are accomplished and governed according to the directed communication network topology. Accordingly, G={V,E,A} is introduced to describe the prescribed topology, where V={1,⋯,N} with V(G)={v1,⋯,vN} and E stands for the sets of nodes and edges, and A=[aij]∈ℝN×N represents the weighted adjacency matrix, respectively. Moreover, by denoting Ni as the local neighbors of sensor i, it can be further assumed that a_ij > 0 when sensor i receives information from sensor j with a_ii = 1 by the ordered pair (i, j), and a_ij = 0 when there is no communication between sensor i and sensor j (Dong et al., 2015). As a result, one can verify that the sensor network is constructed in the fully distributed configuration by using local information exchanges.

Nonlinear dynamical system

Then, consider the following class of continuous-time dynamical systems with nonlinear dynamics and fault signals: (1) {ẋ(t)=Ax(t)+g(x(t))+Ff(t)+Bd(t),x(t)=x0,where x(t)∈ℝn is the dynamical system state, d(t)∈ℝn is the external disturbance belonging to L2[0,∞), g(x(t)) denotes the nonlinearity satisfying Lipschitz conditions, i.e. ||g(α(t))−g(β(t))||≤κ||α(t)−β(t)||, κ>0, ∀α(t),β(t)∈ℝn, f(t)∈ℝl represents the potential fault to be detected, respectively. All the system matrices A, B, F mentioned above are known constant matrices with appropriate dimensions.

Distributed fault estimator design

Without loss of generality, all the sensor nodes within the sensor network communicate with their neighboring nodes according to a unified sampling sequel, which is denoted as tk+1−tk=h(t)>0,k=0,1,2,…. Furthermore, suppose that h(t)≤h¯ can be time-varying with h¯ being a known constant. Based on this distributed sensor network framework, the sampled-data measurement by the ith sensor can be given by: (2) yi(tk)=Cix(tk)+Div(tk),i=1,2,…,N,where yi(tk) stands for the ith sampled-data sensor measurement output, x(tk) implies the sampled-data state of x(t), v(tk) denotes the sampled-data state of v(t) with v(t) being the sensor environment disturbances and C_i, D_i are known system matrix for the sensor nodes.

Subsequently, the distributed fault estimators can be adopted on the ith sensor node: (3) {x^̇i(t)=Ax^i(t)+g(x^i(t))+∑j∈NiaijKij(Cjx^j(tk)−yj(tk)),r^i(t)=∑j∈Niaij(Cjx^j(tk)−yj(tk)),where x^i(t)∈ℝn denotes the state estimation of x(t) by the ith sensor, r^i(t)∈ℝl represents the residual state that can be compatible with f(t), K_ij are the distributed fault estimation gains to be determined later.

Remark 1. It should be pointed out that the distributed fault estimation problem can be well solved by the desired distributed fault estimators, while the residual states on each sensor node in the sensor network can be generated. This implies that each fault estimator can detect the fault and the distributed structure can improve the effectiveness and robustness.

In practical applications of sensor networks, quantified gain uncertainties of the distributed fault estimators should be taken into account in the above model, where ΔKij=M^ijΩ(t)N^ and ΩT(t)Ω(t)≤I, M^ij, and N^ are quantify constant matrices. As a result, the nonfragile design can be achieved with the distributed fault estimator gains.

Then, the distributed fault estimator model can be reformulated as follows: (4) {x^̇i(t)=Ax^i(t)+g(x^i(t))+∑j∈Niaij(Kij+ΔKij)(Cjx^j(tk)−yj(tk)),r^i(t)=∑j∈Niaij(Cjx^j(tk)−yj(tk)),Remark 2. The consideration of nonfragile fault estimator gains is more applicable for practical application on sensor networks, since the designed theoretical results are more robustness with fault estimator implemented uncertainties.

In the following, by letting ei(t)=x^i(t)−x(t) and ri(t)=r^i(t)−f(t), it can be obtained that: (5) {ėi(t)=Aei(t)+G(ei(t))+∑j∈Niaij(Kij+ΔKij)(Cjx^j(tk)−yj(tk))−Ff(t)−Bd(t),ri(t)=∑j∈Niaij(Cjx^j(tk)−yj(tk))−f(t),where G(ei(t))=g(x^i(t))−g(x(t)).

Subsequently, one can further denote that ζ(t)=[xT(t),eT(t)]T with e(t)=[e1T(t),e2T(t),…,eNT(t)]T and w(t)=[dT(t),vT(tk),fT(t)]T with r(t)=[r1T(t),r2T(t),…,rNT(t)]T, such that the following augmented fault estimation error dynamics can be deduced accordingly: (6) {ζ̇(t)=A1ζ(t)+A2ζ(tk)+G(ζ(t))+Bw(t),r(t)=ℂζ(t)+Dw(t),where: A1=[A00(IN⊗A)],A2=[000(K+ΔK)C],K=[aijKij]N×N,ΔK=[aijΔKij]N×N=M^Ω(t)N^=[aijM^ij]N×NΩ(t)N^,B=[B0F−B−(K+ΔK)D−F],B=[BT,BT,…,BT]T,F=[FT,FT,…,FT]T,D=[D1T,D2T,…,DNT]T,C=[0C],C=diag{C1,C2,…,CN},D=[0−D−I],I=[IT,IT,…,IT]T,G(ζ(t))=[g(x(t))G(e(t))],G(e(t))=[gT(x^1(t))−gT(x(t)),gT(x^2(t))−gT(x(t)),…,gT(x^N(t))−gT(x(t))]T.Therefore, the residual evaluation function Ji(t) is confirmed for ith sensor node and the appropriate threshold J_ith is selected for ith sensor node. Moreover, the following logic rule is constructed to cope with the fault detection issues: Jith=sup⁡ω(t)∈L2,f(t)=0∫t0t0+t∗riT(t)ri(t)dt,Ji(t)=∫t0t0+t∗riT(t)ri(t)dt,where t₀ denotes the initial evaluation time instant and t∗ represents the evaluation time length.

Consequently, the occurrence of faults can be detected by comparing J_ith and Ji(t) according to the following relation: Ji(t)>Jith⇒faults occur⇒ generate alarm,Ji(t)≤Jith⇒no faults occur.To this end, the H∞ performance definition is adopted to deal with the external disturbance attenuation during the distributed fault estimation procedure and some useful lemmas are given for deriving the main theoretical results.

Definition 1. Under zero initial conditions, the H∞ performance definition is said to be achieved for system (6), if the formulated distributed fault estimation errors can satisfy the following condition: J∞=∫0∞(rT(t)r(t)−γ2wT(t)w(t))dt<0,where γ is a prescribed performance index.

Lemma 1 (Park et al., 2011). For any matrix M>0, scalars τ>0, τ(t) satisfying 0≤τ(t)≤τ, vector function α̇(t):[−τ,0]→ℝn such that the concerned integrations are well defined, then: −τ∫t−τtα˙T(s)Mα˙(s)ds≤ζT(t)Ωζ(t),where: ζ(t)=[αT(t),αT(t−τ(t)),αT(t−τ)]T,Ω=[−MM0∗−2MM∗∗−M].Lemma 2 (Geromel et al., 1998). For matrix MT=M, and F1, F2 being real matrices of appropriate dimensions with O(t) satisfying OT(t)O(t)≤I, it holds that M+F1O(t)F2+F2TOT(t)F1T<0, if and only if there exists a scalar ε>0 such that M+ε−1F1TF1+εF2TF2<0, or equivalently: [MF1εF2T∗−εI0∗∗−εI]<0.Based on the above discussions, the purpose of our paper is to design the distributed fault estimator gains, so that the fault detection problem can be solved by choosing the appropriate distributed fault detection threshold J_ith with comparison of corresponding residual evaluation function Ji(t) on each sensor nodes.

Main analysis and synthesis results

In this section, the prescribed H∞ performance of distributed fault estimators and the desired fault estimator gain parameters would be analyzed and designed in the form of convex optimization.

Theorem 1. For given h¯>0 and γ>0, the distributed fault detection problem of nonlinear system (1) can be solved with the given fault estimator gains K_i, if there exist symmetric matrices ℙ>0, ℚ>0, ℝ>0, such that the following convex optimization condition: (7) Π=[Π1Π2∗Π3]<0holds with: Π1=[Π11ℙA2+ℝ0∗−2ℝℝ∗∗−ℚ−ℝ],Π11=ℙTA1+A1ℙ+ℚ−ℝ+κ¯I+ℂTℂ,Π2=[ℙℙB+ℂTDhA1Tℝ00hA2Tℝ000],Π3=[−I0hℝ∗−γ2I+DTDhBTℝ∗∗−ℝ].Proof. First, the virtual delay approach can be applied to the augmented fault estimation error system (6), such that one has: (8) {ζ̇(t)=A1ζ(t)+A2ζ(t−τ(t))+G(ζ(t))+Bw(t),r(t)=ℂζ(t)+Dw(t),where 0≤τ(t)=t−tk≤h¯. With the help of virtual delay, the sampled-data dynamical states can be converted to the time-varying delayed states, which can be convenient for later continuous-time dynamical system analysis.

Then, construct the Lyapunov−Krasovskii function for system (8) as follows: (9) V(t)=V1(t)+V2(t)+V3(t),where: V1(t)=ζT(t)ℙζ(t),V2(t)=∫t−h¯tζT(s)ℚζ(s)ds,V3(t)=h¯∫−h¯0∫t+stζ̇T(l)ℝζ̇(l)dlds.Subsequently, along with evolution of V(t), it can be deduced by taking time derivatives that: V̇1(t)=ζ̇T(t)ℙζ(t)+ζT(t)ℙζ̇(t)=2ζT(t)ℙ(A1ζ(t)+A2ζ(tk)+G(ζ(t))+Bw(t)),V̇2(t)=ζT(t)ℚζ(t)−ζT(t−h¯)ℚζ(t−h¯),V̇3(t)=h¯2ζ̇T(t)ℝζ̇(t)−h¯∫t−h¯tζ̇T(s)ℝζ˙(s)ds.Furthermore, it can be verified by Lemma 1 that: −h¯∫t−h¯tζ̇T(s)ℝζ̇(s)ds≤[ζ(t)ζ(t−τ(t))ζ(t−h¯)]T[−ℝℝ0∗−2ℝℝ∗∗−ℝ]×[ζ(t)ζ(t−τ(t))ζ(t−h¯)].In addition, it is noticed that for the formulated nonlinear function G(ζ(t)) with Lipschitz conditions, it holds that: GT(ζ(t))G(ζ(t))≤κ¯ζT(t)ζ(t),where κ¯==diag{κ,κ,…,κ}. Then, the nonlinear dynamical constraints can be represented in the form of linear convex conditions.

Moreover, it can be obtained by the augmented fault estimation error system (6) that: h¯2ζ̇T(t)ℝζ̇(t)=χT(t)[hA1hA20hIhB]ℝ[hA1hA20hIhB]Tχ(t),where χ(t)=[ζT(t),ζT(t−τ(t)),ζT(t−h¯),GT(ζ(t)),wT(t)]T.

Based on the aforementioned derivations, one can compute that: V̇(t)+κ¯ζT(t)ζ(t)−GT(ζ(t))G(ζ(t))+rT(t)r(t)−γ2wT(t)w(t)≤χT(t)(Π¯+[hA1hA20hIhB]ℝ[hA1hA20hIhB]T)χ(t)where: Π¯=[Π¯1ℙA2+ℝ0ℙℙB+ℂTD∗−2ℝℝ00∗∗−ℚ−ℝ00∗∗∗−I0∗∗∗∗−γ2I+DTD],Π¯1=ℙTA1+A1ℙ+ℚ−ℝ+κ¯I+ℂTℂ.Eventually, based on Schur complement lemma for matrix manipulations, it yields that under zero initial conditions if the convex optimization condition Π<0 can be satisfied, then rT(t)r(t)−γ2wT(t)w(t)<0 holds. By integrating formulation from 0 to ∞, it can be obtained that: J∞=∫0∞rT(t)r(t)−γ2wT(t)w(t))dt=∫0∞(rT(t)r(t)−γ2wT(t)w(t)+V̇(t))dt+V(0)−V(∞),such that J∞<0 can be ensured accordingly. Therefore, according to the Definition 1, the augmented fault estimation error system can reach the prescribed H∞ performance and the proof can be immediately completed accordingly. This also means that the distributed fault estimation error dynamics can be well estimated when dealing with the fault estimation problem with the sensor network. □

It is noted that the above derived result in Theorem 1 contains the fault estimator gain fluctuations in the convex optimization problem, which is not a standard convex optimization condition and should be further solved by matrix techniques.

Based on the established condition in Theorem 1, the following theorem is presented to calculate the desired distributed fault estimator gains with the help of linear matrix inequality optimization.

Theorem 2. For given h¯>0 and γ>0, the distributed fault detection problem of nonlinear system (1) can be solved, if there exist symmetric matrices ℙ=diag{P1,P2}>0 with P2=diag{P2,P3,…,PN+1}, ℚ=diag{Q1,Q2}>0, ℝ=diag{R1,R2}>0, and matrix W, such that the following convex optimization condition: (10) Ξ=[Ξ1Ξ2∗Ξ3]<0holds with: Ξ1=[Ξ11Ξ12∗Ξ13],Ξ11=[Ξ1110R100∗Ξ1120WC+R20∗∗−2R10R1∗∗∗−2R20∗∗∗∗−Q1−R1],Ξ111=2P1A+Q1−R1+κ¯I,Ξ112=2P2(IN⊗A)+Q2−R2+κ¯I+CTC,Ξ12=[0P10P1B000P2−P2B−WD−CTD00000R2000000000],Ξ13=[−Q2−R20000∗−I000∗∗−I00∗∗∗−γ2I0∗∗∗∗−γ2I+DTD],Ξ2=[Ξ21,Ξ22],Ξ21=[P1FhATP1−P2F−CTI0000000000hP1000hBTP1DTI0],Ξ22=[000h(IN⊗A)TP2P2M^0000hCTWT0εCTN^T000000000hP200−hBTP200−hDTWT0−εDTN^T],Ξ3=[−γ2I+ITIhFTP1−hFTP200∗P1−2R1000∗∗P2−2R2hP2M^0∗∗∗−εI0∗∗∗∗−εI].With above feasible solutions, the desired fault estimator gains K_ij, i,j=1,2,…N, can be obtained by sensor network topology: K=P2−1W,Proof. By performing congruent transformation to the derived convex optimization condition in Theorem 1 and recalling the augmented matrices in the system (6), one has that: Ξ¯+XΩ(t)Y+YTΩ(t)XT<0,where: Ξ¯=[Ξ¯1Ξ¯2∗Ξ¯3],Ξ¯1=[Ξ¯11Ξ¯12∗Ξ¯13],Ξ¯11=[2P1A+Q1−R1+κ¯I0R1∗Ξ¯1110∗∗−2R1],Ξ¯111=2P2(IN⊗A)+Q2−R2+κ¯I+CTCΞ¯12=[000P10P1BP2KC+R2000P2−P2B0R10000],Ξ¯13=[Ξ¯1310∗Ξ¯132],Ξ¯131=[−2R20R2∗−Q1−R10∗∗−Q2−R2],Ξ¯132=[−I00∗−I0∗∗−γ2I],Ξ¯2=[Ξ¯21,Ξ¯22],Ξ¯21=[0P1F−P2KD−CTD−P2F−CTI00000000000000],Ξ¯22=[hATP100h(IN⊗A)TP2000hCTKTP20000hP100hP2hBTP1−hBTP2],Ξ¯3=[Ξ¯31Ξ¯32∗Ξ¯33],Ξ¯31=[−γ2I+DTDDTI∗−γ2I+ITI],Ξ¯32=[0−hDTKTP2hFTP1−hFTP2],Ξ¯33=[P1−2R10∗P2−2R2].and: X=[0P2M^0000000000hP2M^],Y=[000N^C00000−N^D000].Consequently, it is denoted that W=P2K, such that the following convex optimization condition Ξ<0 can be further derived based on Lemma 2 and the rest of proof can follow directly from that in Proof 1. Eventually, the nonfragile design framework can be achieved by using the convex matrix transformation and the corresponding implementation robustness can be ensured accordingly.

Remark 3. It is worth mentioning that our developed convex optimization conditions are presented in the form of strict linear matrix inequality by feasible matrix solutions. Furthermore, the optimized H∞ performance index γ can be computed by: min⁡γ,s.t.Ξ<0.The corresponding computation complexity is determined by the number of sensor nodes and the dimension of dynamical system states. This means that when adopting the practical sensor network, the structure and topology should be considered with the fault estimator design and nonfragile framework to acquire the efficiency and robustness.

In all, the following algorithm can be represented based on the above theoretical results.

Algorithm 1 The fault estimator gains calculation based on sensor networks1:  Step 1: Given H∞ performance index γ and sensornetwork topology structure, select the initial matrixvalues for the nonlinear dynamical system,2:  Step 2: Given sampling period h¯ for the sensormeasurement output.3:  Step 3: Solving the convex optimization condition inTheorem 2.4:  Step 4: Derive the distributed estimator parametermatrices according to the sensor network topologystructure.5:  Step 5: Stop.

Simulation example

In this section, a simulation example is used to demonstrate the effectiveness and advantage of our proposed nonfragile distributed fault estimation method for the nonlinear dynamical system in sensor networks.

In the simulation, consider a nonlinear dynamical system as (1) with the following system parameters: A=[−30.10.5−2.5],g(x(t))=[0.5tanh⁡(x1)−0.5tanh⁡(x2)],F=[0.5−0.4],B=[0.20.3].The potential fault signal is supposed to be a square ware signal with f(t) = 1, 2s≤t≤5s, while the external disturbances for the sensor nodes are given by d(t)=0.2sin⁡(t) and v(t)=0.1cos⁡(t).

To estimate the fault signal, a distributed sensor network of four nodes in the simulation can be adopted with following parameters: C1=[1.051.05],C2=[11],C3=[0.950.95],C4=[0.90.9],and: D1=0.2,D2=0.3,D3=0.4,D4=0.5.The set of communication network edges E are given by {(1,1),(1,2),(2,2),(2,3),(3,1),(3,3),(4,2),(4,4)} and the corresponding communication network topology matrix is set by: A=[1100011010100101].Based on our proposed distributed fault estimators with nonfragile consideration, the state estimator gain fluctuation is assumed as: M^ij=[0.1000.2],i,j=1,2,3,4.and: N^=[0.5000.4].Thus, by solving the convex optimization condition in Theorem 2 with the above simulation parameters, the desired fault estimator gains can be obtained by solving the linear matrix inequities as: K11=[−3.7722−3.1746],K12=[−0.11000.4396],K22=[−3.6499−3.1077],K23=[0.20940.8027],K31=[0.14340.5424],K33=[−3.2845−2.6091],K42=[0.39870.7587],K44=[−3.0048−2.3074].In the simulation, the prescribed H∞ performance is set as 2.5 and the random initial conditions are used. The simulation is performed with Matlab/Simulink software. As a result, the fault signal can be depicted in Figure 1. As a result, Figures 2–5 shows the evolution of residual function Ji(t),i=1,2,3,4 for the sensor network with both cases with fault and fault-free, respectively. One can see that the evolution of residual function dynamics can be well generated by the fault state estimator design. By mathematical calculation, it can be obtained that for t∈[0,30], the fault threshold can be determined by each J_ith for sensor nodes, respectively. Consequently, the appeared fault can be detected after it occurs and the alarm signal would be generated accordingly, which can well support our proposed fault estimation approach and the external disturbances can be attenuated to a desired H∞ performance level.

Conclusion and prospect

In this paper, the distributed fault estimation issue for a class of nonlinear dynamical systems is discussed based on the set of sensor networks. More precisely, the group of fault estimators are designed with consideration of fully distributed communication manner. In addition, the fault estimator gain fluctuations are introduced according to the nonfragile practical applications. Then, sufficient convex optimization criteria are derived in terms of Lyapunov−Krasovskii method, such that the desired H∞ performance can be satisfied and the corresponding distributed estimator gains can be designed. Finally, the simulation example is presented with theoretical validation details. In our future research, an interesting topic is focused on the more realistic sensor networks constraints, especially for the affection of network resource limitations.

Author Affiliation

Figure 1

The fault signal dynamics

Figure 2

The residual evolution function dynamics of sensor 1

Figure 3

The residual evolution function dynamics of sensor 2

Figure 4

The residual evolution function dynamics of sensor 3

Figure 5

The residual evolution function dynamics of sensor 4

This work was supported by the National Natural Science Foundation of China under Grant 52174352.

Conflict of interest: The authors declare that there are no conflict of interest.

References

Akyildiz, I.F., Su, W., Sankarasubramaniam, Y. and Cayirci, E. (2002), “A survey on sensor networks”, IEEE Communications Magazine, Vol. 40 No. 8, pp. 102-114.

Chen, C., Zhu, S., Guan, X. and Shen, X. (2014), “Consensus-based optimal target tracking in relay assisted wireless sensor networks”, Wireless Sensor Networks: Distributed Consensus Estimation, Springer International Publishing, Cham, pp. 41-64.

Chen, L., Zhu, Y., Wu, F. and Zhao, Y. (2023), “Fault estimation observer design for Markovian jump systems with nondifferentiable actuator and sensor failures”, IEEE Transactions on Cybernetics, Vol. 53 No. 6, pp. 3844-3858.

Dong, H., Wang, Z., Ding, S.X. and Gao, H. (2014), “A survey on distributed filtering and fault detection for sensor networks”, Mathematical Problems in Engineering, Vol. 2014 No. 1, p. 858624.

Dong, H., Wang, Z., Alsaadi, F.E. and Ahmad, B. (2015), “Event-triggered robust distributed state estimation for sensor networks with state-dependent noises”, International Journal of General Systems, Vol. 44 No. 2, pp. 254-266.

Dong, H., Wang, Z., Bu, X. and Alsaadi, F.E (2016), “Distributed fault estimation with randomly occurring uncertainties over sensor networks”, International Journal of General Systems, Vol. 45 No. 5, pp. 662-674.

Elhabyan, R., Shi, W. and St-Hilaire, M. (2019), “Coverage protocols for wireless sensor networks: review and future directions”, Journal of Communications and Networks, Vol. 21 No. 1, pp. 45-60.

Geromel, J., de Oliveira, M.C. and Hsu, L. (1998), “LMI characterization of structural and robust stability”, Linear Algebra and Its Applications, Vol. 285 Nos 1/3, pp. 69-80.

Gungor, V.C., Lu, B. and Hancke, G.P. (2010), “Opportunities and challenges of wireless sensor networks in smart grid”, IEEE Transactions on Industrial Electronics, Vol. 57 No. 10, pp. 3557-3564.

Isermann, R. (1984), “Process fault detection based on modeling and estimation methodsła survey”, Automatica, Vol. 20 No. 4, pp. 387-404.

Isermann, R. (1997), “Supervision, fault-detection and fault-diagnosis methodsłan introduction”, Control Engineering Practice, Vol. 5 No. 5, pp. 639-652.

Kaplan, L.M. (2006), “Local node selection for localization in a distributed sensor network”, IEEE Transactions on Aerospace and Electronic Systems, Vol. 42 No. 1, pp. 136-146.

Kumar, P. and Lee, H.J. (2011), “Security issues in healthcare applications using wireless medical sensor networks: a survey”, Sensors, Vol. 12 No. 1, pp. 55-91.

Lan, Y.H. and Zhou, Y. (2013), “Non-fragile observer-based robust control for a class of fractional-order nonlinear systems”, Systems & Control Letters, Vol. 62 No. 12, pp. 1143-1150.

Lan, J. and Patton, R.J. (2016), “A new strategy for integration of fault estimation within fault-tolerant control”, Automatica, Vol. 69, pp. 48-59.

Leung, H., Chandana, S. and Wei, S. (2008), “Distributed sensing based on intelligent sensor networks”, IEEE Circuits and Systems Magazine, Vol. 8 No. 2, pp. 38-52.

Li, H., Gao, Y., Shi, P. and Lam, H.-K. (2015), “Observer-based fault detection for nonlinear systems with sensor fault and limited communication capacity”, IEEE Transactions on Automatic Control, Vol. 61 No. 9, pp. 2745-2751.

Li, H., Chen, Z., Wu, L. and Lam, H.-K. (2016), “Event-triggered fault detection of nonlinear networked systems”, IEEE Transactions on Cybernetics, Vol. 47 No. 4, pp. 1041-1052.

Li, J., Dong, H., Han, F. and Li, X. (2017), “Filter design, fault estimation and reliable control for networked time-varying systems: a survey”, Systems Science & Control Engineering, Vol. 5 No. 1, pp. 331-341.

Muhammed, T. and Shaikh, R.A. (2017), “An analysis of fault detection strategies in wireless sensor networks”, Journal of Network and Computer Applications, Vol. 78, pp. 267-287.

Ota, N. and Wright, P. (2006), “Trends in wireless sensor networks for manufacturing”, International Journal of Manufacturing Research, Vol. 1 No. 1, pp. 3-17.

Park, P., Ko, J.W. and Jeong, C. (2011), “Reciprocally convex approach to stability of systems with time-varying delays”, Automatica, Vol. 47 No. 1, pp. 235-238.

Qi, H., Iyengar, S.S. and Chakrabarty, K. (2001), “Distributed sensor networksła review of recent research”, Journal of the Franklin Institute, Vol. 338 No. 6, pp. 655-668.

Shahrampour, S., Rakhlin, A. and Jadbabaie, A. (2015), “Distributed detection: finite-time analysis and impact of network topology”, IEEE Transactions on Automatic Control, Vol. 61 No. 11, pp. 3256-3268.

Wang, D., Liu, J., Xu, J. and Wang, C. (2015), “Data sweeper: a proactive filtering framework for error-bounded sensor data collection”, IEEE Transactions on Emerging Topics in Computing, Vol. 4 No. 4, pp. 487-501.

Xia, J., Gao, H., Liu, M. and Zhang, B. (2018), “Non-fragile finite-time extended dissipative control for a class of uncertain discrete time switched linear systems”, Journal of the Franklin Institute, Vol. 355 No. 6, pp. 3031-3049.

Xu, N., Zhu, Y., Yang, R. and Su, C.-Y. (2023), “Adaptive fault-tolerant control for a 2-body point absorber wave energy converter against actuator faults: an iterative learning control approach”, IEEE Transactions on Sustainable Energy, Vol. 14 No. 3, pp. 1664-1675, doi: 10.1109/TSTE.2023.3243030.

Yang, Y., Ding, S.X. and Li, L. (2015), “On observer-based fault detection for nonlinear systems”, Systems & Control Letters, Vol. 82, pp. 18-25.

Yang, H., Zhang, J., Jia, X. and Li, S. (2019), “Non-fragile control of positive Markovian jump systems”, Journal of the Franklin Institute, Vol. 356 No. 5, pp. 2742-2758.

Zhang, K., Jiang, B. and Shi, P. (2009), “Fast fault estimation and accommodation for dynamical systems”, IET Control Theory & Applications, Vol. 3 No. 2, pp. 189-199.

Zhu, Y. and Zheng, W.X. (2023), “An integrated design approach for fault-tolerant control of switched LPV systems with actuator faults”, IEEE Transactions on Systems, Man, and Cybernetics: Systems, Vol. 53 No. 2, pp. 908-921.

Zhu, S., Chen, C., Li, W. and Guan, X. (2013), “Distributed optimal consensus filter for target tracking in heterogeneous sensor networks”, IEEE Transactions on Cybernetics, Vol. 43 No. 6, pp. 1963-1976.