Set-Membership Filtering Approach to Dynamic Event-Triggered Fault Estimation for a Class of Nonlinear Time-Varying Complex Networks

Xiaoting Du , Lei Zou ,,, and Maiying Zhong

Abstract—The present study addresses the problem of fault estimation for a specific class of nonlinear time-varying complex networks, utilizing an unknown-input-observer approach within the framework of dynamic event-triggered mechanism (DETM).In order to optimize communication resource utilization, the DETM is employed to determine whether the current measurement data should be transmitted to the estimator or not.To guarantee a satisfactory estimation performance for the fault signal,an unknown-input-observer-based estimator is constructed to decouple the estimation error dynamics from the influence of fault signals.The aim of this paper is to find the suitable estimator parameters under the effects of DETM such that both the state estimates and fault estimates are confined within two sets of closed ellipsoid domains.The techniques of recursive matrix inequality are applied to derive sufficient conditions for the existence of the desired estimator, ensuring that the specified performance requirements are met under certain conditions.Then, the estimator gains are derived by minimizing the ellipsoid domain in the sense of trace and a recursive estimator parameter design algorithm is then provided.Finally, a numerical example is conducted to demonstrate the effectiveness of the designed estimator.

I.INTRODUCTION

COMPLEX networks (CNs) are a special kind of systems that are composed of a series of interconnected nodes which can be regarded as the basic units with their distinct dynamical behaviors.The examples of CNs in practical applications include the well-known world wide web, genetic networks, power grid networks, and social networks.The analysis of CNs needs to consider the dynamic characteristics of individual nodes as well as the coupling configurations between nodes [1]-[5].In recent decades, emphasis has been placed on problems related to the dynamic analysis of CNs,such as synchronization [6], state estimation [7], and extension mechanisms.On the other hand, it should be pointed out that fault diagnosis represents another significant dynamic analysis issue due to the wide existence of fault signals, which has then promoted the study of fault diagnosis for CNs.

Until now, the problems of fault diagnosis for CNs have garnered significant research focus [8]-[12].For example, the problem of fault detection for networked dynamical systems has been examined in [8].The issue of fault diagnosis for time-varying systems over sensor networks using the Round-Robin protocol has been addressed in [9].The issue of fault estimation and accommodation for interconnected systems has been discussed in [12] by employing a separation principle.Among various research results reported in the literature, fault estimation plays a pivotal role in fault diagnosis issues due to its capacity of providing the estimates on the shape and size for the occurred fault signals.It should be noted that there are different techniques (e.g., adaptive-observer-based method,sliding-mode-observer-based approach, etc.) dealing with various kinds of estimation problems (including the state estimation, parameter estimation, and fault estimation).Among these techniques, the essential idea of unknown-input-observer(UIO)-based technique is to derive the desired estimate by effectively separating the unwanted uncertainties/disturbances from the estimation process, thereby minimizing/eliminating their impacts on estimation performance.In contrast to other types of estimators, the UIO-based estimator [13] could ensure a satisfactory estimation performance by eliminating the effects from the unknown faults without any priori assumptions about fault information, and thereby theoretically easier engineering implementation of the estimation of mutation-like faults.Unfortunately, due to the mathematical complexity, the UIO-based fault estimation problem (FEP) for CNs has not received sufficient attention which is the primary motivation for this paper.

Accuracy is one of the most essential performance indices of estimation problems (including the state estimation and fault estimation).To date, there are different criteria available in the literature for quantifying the accuracy of the estimation issues subject to different kinds of disturbances/noises (such as Gaussian noises, energy-bounded noises, and unknown but bounded (UBB) noises).It is worth mentioning that most of the existing results concerning the estimation problems of CNs have drawn their attention on the Gaussian noises and energy-bounded noises, with the minimum-variance index and the traditional H∞index fully exploited.Nevertheless, when it comes to the CNs with UBB noises, the corresponding research results are rare.

Set-membership filtering (SMF) scheme is an effective method dealing the estimation issue subject to UBB noises.Such an approach was first introduced in [14], with the main idea of obtaining a confidence region that encompasses the true states of the targets.Unlike point-wise state estimation techniques (e.g., H∞filtering and Kalman/extended Kalman approaches), the SMF method is a typical interval estimation strategy with complementary regions that provides sufficient information to ensure safe and reliable monitoring boundary for the targets, thus contributing to the reliability of real industrial systems, even in the face of unexpected environmental changes.A significant body of research studies on SMF issues has been published in the literatures to date [15].

Over the past two decades, there has been a significant surge in research interest regarding event-based filtering/control problems for various systems [16]-[19].The utilization of the event-triggered mechanism (ETM) would significantly minimize the superfluous consumption of network resources,thereby contributing to the relieving of network congestion[18], [19].In recent years, significant efforts have been made in the study of DETMs, with a focus on dynamically adjusting the threshold value (or threshold parameter) of the triggering function based on system evolution [20]-[24].For example, in [22], the fault-tolerant control and FEPs have been investigated for networked control systems under the scheduling of DETM.The study in [23] has focused on fault estimation and fault-tolerant control for a class of continuous-time dynamic systems.To reduce the amount of data transfer, internal dynamic variables were introduced in the ETM, which enlarged the inter-event interval.The design of a DETM has been proposed in [24], utilizing information from the system output and the observer function.Additionally, the problem of designing a sliding mode observer has been addressed specifically for uncertain fuzzy time-delay systems.

Building upon the preceding discussion, we aim to study the dynamic event-triggered FEP of a class of nonlinear timevarying CNs subject to UBB noises.The following three fundamental challenges have subsequently been identified: 1) How to handle the coupling effects of state vector and fault vector on the resultant estimation performance? 2) How to handle the effects induced by dynamic event-triggered mechanism on the estimator design? and 3) How to design the time-varying parameter for the estimator such that the corresponding estimation algorithm is propitious to online operation?

The primary contributions of this paper are emphasized as follows for the identified challenges.1) For the first time, FEP is investigated for nonlinear time-varying CNs with UBB noises under the effects of DETM; 2) A non-fragile event-triggering condition has been introduced to describe the rounding errors arisen during digital implementation; 3) A novel fault estimator, based on the UIO, is designed to ensure optimal performance in fault estimation by leveraging the decoupling technique; 4) The desired parameters for the fault estimator are recursively obtained by solving a set of matrix inequalities, thereby facilitating their real-time application.

The remainder of this paper is structured as follows.In Section II, the FEP for CNs is formulated using the DETM.The design scheme of the desired observer is provided in Section III.The effectiveness of the established methods is illustrated through a simulation example in Section IV, and the conclusion is derived in Section V.

II.PROBLEM FORMULATION AND PRELIMINARIES

Consider a class of discrete time-varying CNs with N coupled nodes as follows:

Assumption 1: The noise ω(s) and ν(s) are confined to the ellipsoidal sets described below, respectively;

where ? (s) and ? (s) are known positive definite matrices.

Assumption 2: The initial statex(0) and initial faultf(0) satisfy the following conditions:

wherex?(0) represents the estimation of the initial statex(0),f?(0) stands for the estimation of the initial faultf(0), P (0) and G(0)are given positive definite matrices.

Assumption 3: The nonlinear functionhˉ(·) satisfies sectorbounded condition

whereU1(s) andU2(s) are known matrices with appropriate dimensions andhˉ(0)=0.

Assumption 4: For any integers≥0, the matrices Ri(s) and Ci(s) satisfy r ank(Ri(s))=rank(Ci(s)Ri(s)),rank(Ri(s)TCi(s)T,I)=rank(Ri(s)TCi(s)T) for alli=1,2,...,N.Here,Istands for identity matrix.

Remark 1: The existence of the UIO is guaranteed by Assumption 4, as demonstrated in [25].In fact, Assumption 4 demonstrates that part of the measurement outputs are irrelevant to the external inputs (i.e., the fault signals).In practical applications, it is frequently encountered that only some of the measurement outputs are impacted by fault signals, which demonstrates the reasonability of this assumption.

The signal transmission between the fault estimator and the device is accomplished through a communication network with restricted bandwidth in this paper.A reasonable signal triggered mechanism would contribute to the improvement of the utilization efficiency for the communication resource.ETMs, as opposed to traditional time-based triggered mechanisms, are a non-equal-cycle “on-demand” scheme that effectively improves resource utilization while maintaining system performance.To further alleviate the network communication load, an enhanced ETM (namely, DETM) is utilized to handle signal transmissions between the estimator and sensors, which determines whether or not to transmit the current system outputyi(s) over the communication channels.More specifically,the signal transmissions occur only when the triggering condition described below is satisfied:

Remark 2: The traditional static-triggering condition is obtained from the dynamic-triggering condition (5) when the parameter θiapproaches infinity, as shown by (6).The traditional static-triggered mechanism can be considered as a specific case of the DETM, in other terms.The total number of triggering occurrences can be minimized by optimizing three parameters (i.e., θi, σiand λi), thereby giving rise to a better flexibility as compared to the static-triggered mechanism [26].

For convenience, the following notations are defined:

According to the above definitions, we rewrite the dynamics of the CNs (1) as follows:

Similarly, the UIO-based fault estimator can be rewritten as follows:

where A ︿(s)?A(s)+W ??！? V ˉ(s)?V(s)+W ??！?

Denoting the estimation error of the state ase(s)?x(s)-x?(s), and subtracting (10) from (9), we have

To simplify the calculation of the above error equation, first,let’s adopt the UIO design concept in [27] to redefine the matrix K (s) as follows:

On the other hand, it is observed that

in which ?(s)?[?T1(s) ?T2(s) ··· ?TN(s)]T.Then, substituting(12) and (13) into (11) yields

Furthermore, it can be derived from (9) that

Considering the dynamical behavior shown in (15), it is observed that the state estimation error would be affected through the fault signal if J (s+1)C(s+1)R(s)≠R(s).To ensure a satisfactory estimation performance, we can decouple the estimation error from the effects of the unknown signal by designing the parameter J(s+1) such thatJ(s+1)C(s+1)E(s)=E(s).Moreover, by designingT(s)=I-J(s+1)C(s+1), the estimation error dynamics (15) can be rewritten as follows:

According to Assumption 3, it is easy to see that the effects ofcan be roughly determined by the estimation errore(s).Hence, it is concluded from (16) that the dynamics of estimation error can be roughly determined by the noises, the initial statee(0) and the triggering error ?(s),which eliminates the impacts of the unknown fault signals.

Following the dynamics (16), the fault estimation for the system (9) can be generated based on the state estimation error dynamics (16).From (7), we have

Along the similar lines in the design of J(s+1), let’s design the matrix H(s) such that H(s)C(s+1)R(s)=I.Then,the above equation can be rewritten as

Subtractingf(s) from both sides of the equation and denoting the estimation error of fault as ～f(s)?f?(s)-f(s), one can infer that

which completes the design of the estimator parameter H (s).

The schematic diagram of the filtering system under consideration in this paper is depicted in Fig.1.The aim of this paper is to develop a set-membership filter such that, under the event-triggered communication mechanism, the target state errore(s+1) and fault error ～f(s) lie in the following closed ellipsoid domains:

where P(s+1) and G(s+1) stand for positive definite matrices to be determined.Without any loss of generality, the condition (21) is referred to as (P(s),G(s))-dependent constraint in this paper.

III.MAIN RESULT

Fig.1.Schematic structure for the plant and the filter over a network (with the DET protocol).

The filter parameters for CN (9) will be designed in this section, based on the output informationy?(s).First, we shall consider the existence condition for the filter (10) in order to ensure that the filtering error system (16) satisfies the constraint (21) that is dependent of ( P(s),G(s)).

Along the similar lines of UIO design technique [27], the estimator parameters should be designed such that the following constraints are satisfied:

The state error estimation (16) is then simplified to

where Vˉ(s), T(s), K1(s), J(s+1) are parameters to be designed.

The previous discussions clearly indicate that the condition R(s)=J(s+1)C(s+1)E(s)holds under Assumption 4 by the following designing:

or

Lemma 2(S-Procedure Lemma[28]): Let π0(·),π1(·),...,

which means that

where

Finally, Lemma 2 shows that if there are positive scalars ??=1,2,...,7(s)such that

So far, we have analyzed the closed ellipsoid domains for fault estimation error and the state estimation error in Theorem 1 using mathematical induction.Theorem 1 states that,given an initial filtering error matrix P(0) and matrices ?(s),?(s), ?(s+1) which represent the limitations imposed by external noises, it is always possible to identify an ellipsoid that encompasses a trajectory of the filtering errore(s).

The subsequent step involves the design of the filter gain matrix to achieve optimal filtering performance by minimizing the size of the constraint ellipsoid, as stated in the forthcoming theorem.The filter gain matrices K(s) are then determined recursively based on the derived time-varying parameter K1(s).

Theorem 2: Considering the CNs (9), assume that for anys≥0, there exist estimator gain matrix K1(s), positive definite matrix P(s+1) and positive scalars??(s)(?=1,2,...,7)such that the matrix inequality (28) is satisfied.Then, the size of the ellipsoid constraint P(s) can be minimized by solving the following constrained optimization problem:

subject to the constraint (28).

Proof: The proof is self-evident and, for the sake of brevity,is omitted here.■

Algorithm 1 to determine the filter gain matrices is outlined as follows, based on Theorem 2

Remark 4: Our main findings are derived based on the linear matrix inequality (LMI), and the corresponding algorithm exhibits a polynomial time complexity.Specifically, let M

Algorithm 1 Computational Algorithm for Theorem 2 Step 1: Initialization: Set.Give the maximum simulation times, matrix which satisfies (3) and matrices , ,which satisfy Assumption 1.s=0 smax P(0) ?(s) ?(s)?(s+1)ˉE(s) P(s)Step 2: Calculate Cholesky factorization of.Step 3: Solve the optimization problem (49) while considering matrix inequality constraints (28).Then, the filter parameters and shape matrix can be obtained.s=s+1 s ≤smax K1(s) P(s+1)Step 4: Set.If , proceed to Step 2 otherwise exit.

IV.AN ILLUSTRATIVE EXAMPLE

The accuracy and effectiveness of our developed estimator approach are demonstrated through a numerical example in this section.

Consider the CN described in (9) with the following parameters:

Suppose that the coupled configuration is of the form of N ={1,2,3}

The noises are assumed to be ωi(s)=0.1sin(0.5s) and νi(s)=0.1sin(0.5s), respectively.The initial value of the internal dynamic variable for the dynamic triggering conditions (5) and (6) is set to be η (0)=0, while the threshold is set to be σi(s)=0.01.The additional parameters are chosen as θi(s)=20 and λi(s)=0.1.

The nonlinear vector-valued functions are provided as

with the below instantly obtained parameters:

The simulation incorporates the subsequent segmented fault

The trajectories of state, fault and their estimates are shown in Figs.2-6, which demonstrate the validity of our proposed approach.The introduction of the DETM in Fig.7 results in a reduction in information transmission and effective conservation of network resources.

Fig.2.The state trajectories of x1s and x?1s.

Fig.3.The state trajectories of x2s and x?2s.

V.CONCLUSION

Fig.4.The error trajectories of xs and x?s.

Fig.5.The fault trajectories of fs and f?s.

Fig.6.The error trajectory of fs and f?s.

In this paper, the UIO-based FEP has been addressed for a class of nonlinear time-varying CNs under the DETM.A discrete-time version DETM has been proposed to save communication resource.The UIO method has been utilized to construct a fault estimator.By adopting the set-membership filtering approach, sufficient conditions have been established for the desired estimator to confine both the state estimates and fault estimates within two sets of closed ellipsoid domains.The desired gains for the estimator have been computed through solving a sequence of optimization problems subject to constraints.Finally, a numerical example has been given to illustrate the correctness and effectiveness of the proposed fault estimation method.Future research topics include 1) The communication-protocol-based fault estimation for CNs [29]-[32], and 2) The fault estimation for CNs with encodingdecoding mechanism [33], [34].

Fig.7.The dynamic triggering instants for estimator.