Event-triggered fault detection for discrete-time Lipschitz nonlinear networked systems in finite-frequency domain

Communicated by Dr Hu Jun

Accepted Manuscript

Event-triggered fault detection for discrete-time Lipschitz nonlinear networked systems in finite-frequency domain Ying Gu, Guang-Hong Yang PII: DOI: Reference:

S0925-2312(17)30746-4 10.1016/j.neucom.2017.04.037 NEUCOM 18370

To appear in:

Neurocomputing

Received date: Revised date: Accepted date:

20 January 2017 13 April 2017 24 April 2017

Please cite this article as: Ying Gu, Guang-Hong Yang, Event-triggered fault detection for discretetime Lipschitz nonlinear networked systems in finite-frequency domain, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.04.037

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Event-triggered fault detection for discrete-time Lipschitz nonlinear networked systems in finite-frequency domain Ying Gua,b , Guang-Hong Yanga,c,∗ a

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College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, 110819, P. R. China b College of Science, Dalian Jiaotong University, Dalian, Liaoning, 116028, P. R. China c State Key Laboratory of Synthetical Automation of Process Industries, Northeastern University, Shenyang, Liaoning, 110819, P. R. China

Abstract

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The problem of event-triggered fault detection (FD) filter design for discretetime Lipschitz nonlinear networked systems with finite-frequency specifications is investigated. The event-triggered transmission scheme is introduced to mitigate the utility of limited network bandwidth. The developed filter combines the H∞ and H− indices. For this class of systems, the generalized Kalman-Yakubovic-Popov lemma-based finite-frequency FD filter design methods are invalid. To solve this problem, the nonlinear error dynamics are transformed into a linear parameter varying (LPV) system based on the use of a reformulated Lipschitz property and a new lemma is developed to capture the system performances in finite-frequency domain. By introducing slack variable techniques, sufficient conditions for the design of FD filter are derived in terms of linear matrix inequalities (LMIs). The proposed design method can significantly reduce the data transmission and achieve a better FD performance than that in full frequency domain. Finally, two examples are presented to validate the effectiveness and superiority of the new results.

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Keywords: Networked control systems, Lipschitz nonlinear systems, Fault detection, Event-triggered scheme, Finite frequency, H− /H∞ . ∗

Corresponding author Email addresses: [email protected] (Ying Gu), [email protected] (Guang-Hong Yang) Preprint submitted to Neurocomputing

April 29, 2017

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1. Introduction

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Networked control systems (NCSs) have received a great deal of attention due to the rapid development of communication technology. The main advantages of NCSs are low cost, simple installation and maintenance, high efficiency and reliability. As a result, NCSs are successfully applicable to many fields, ranging from mobile communication, advanced aircraft, remote surgery and manufacturing processes. However, since the network cable is of limited capacity, the insertion of network brings about new challenging issues including transmission delay, signal quantisation, data missing, etc. Recently, extensive efforts have been made on estimation, analysis, and controller synthesis for NCSs, see, for example [1, 2, 3, 4, 5] and the references therein. Practically all the control tasks executed today are based on digital sampling measurements. The simplest approach is to transmit measurements periodically. Although periodic sampling is preferred for system modeling and analysis, the limited communication band and inadequate computation resources are the problems that often have to be dealt with in a network environment. Under typical periodic sampling, a lot of unnecessary sampled data is transmitted even when the output fluctuation is small compared with the previous one, which may lead to excessive utilisation of the limited bands. Therefore, how to reduce data transmission while guaranteeing the desired performances of NCSs is an important topic. Recently, event-triggered data transmission scheme has been proposed for its capacity of reducing the usage of communication resources in NCSs. There is an increasing interest in using event-triggered strategy for control [6, 7, 8, 9] and filtering [10, 11, 12, 13] areas. Event-triggered method advocates data transmission only when there is a significant change in the signal. Therefore, the burden of the network communication is reduced. Nonlinearities ubiquitously exist in practical systems. As most real physical processes can be described by Lipschitz nonlinear models, considerable interest has been shown on designing observers or filters for Lipschitz systems. Many important results have been reported in the literature, see for example [14, 15, 16]. For Lipschitz nonlinear networked control systems, fruitful results have been obtained with the consideration of different network-induced phenomena [17, 18, 19, 20]. However, these results are all based on the traditional time-triggered communication scheme, which is not efficient for saving the network bandwidth. Besides, the provided synthesis conditions 2

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in all these methods are generally infeasible for systems with large Lipschitz constants. A reformulated Lipschitz property proposed in [21] can be used to cope with this restriction. Due to the introduction of a less conservative Lipschitz condition, the proposed design method can provide less restrictive synthesis conditions than those reported in the literature and avoids high gain. Fault detection (FD), on the other hand, has received much attention due to the increasing demand for reliability and safety in industrial process. A lot of detection methods have been achieved, for instance, see [22, 23] and the references therein. Among these FD methods, robust FD scheme is accepted as a useful and popular one. The main challenge in robust FD is to generate a residual signal that should be robust to the disturbances and simultaneously sensitive to the faults. Then compare the residual signal with a predefined threshold. The robustness and sensitivity should be measured by a suitable performance index and be optimized, e.g., H∞ optimization [24], H2 /H∞ optimization [25] or mixed H− /H∞ optimization [26, 27]. The idea of event-based control can be applicable for FD purposes. In [28], the event-based FD for the networked systems with communication delay and nonlinear perturbation has been investigated. Recently, the problem of eventtriggered fault detection and isolation filter design for discrete-time linear systems was addressed in [29]. [30] studied the problems of event-triggered multi-objective synthesis of feedback controllers and fault diagnosis filters in a unified framework for discrete-time linear systems. [31] investigated fuzzy FD problem under event-triggered scheme. [32] studied the problem of event-triggered FD filter and controller coordinated design for continuoustime NCSs with biased sensor faults. Furthermore, it is worth noticing that all the aforementioned FD approaches were considered in full frequency domain. However, in practice, the frequency ranges of external disturbances and faults may be known beforehand. For instance, incipient faults emerge in low frequency domain as the fault development is slow [22]. Therefore, it will be conservative to design full frequency filters to detect faults due to the overdesign. To cope with the frequency of the signals, the generalized Kalman-Yakubovich-Popov (GKYP) lemma [33] based finite-frequency H∞ control and filtering problems have been studied in [34, 35] for T-S fuzzy systems. More recently, finite frequency FD methods have been presented in [36, 37, 38]. However, these existing finite-frequency FD results are invalid for nonlinear networked control systems with Lipschitz nonlinearity under event-triggered scheme. To 3

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the best of our knowledge, the FD problem for Lipschitz nonlinear systems in finite-frequency domain has not been investigated yet, not to mention the case when event-triggered scheme is considered. Therefore, the main purpose of this paper is to shorten such a gap. The main contributions in this paper are as follows: (1) A reformulation of the Lipschitz property is considered to transform the nonlinear error dynamics into a LPV system [21]. (2) An H− /H∞ FD method with finite-frequency specifications under event-triggered scheme is developed, and a novel lemma is derived to guarantee the whole system with desired performances in finitefrequency domain. (3) By using slack variable techniques, the non-convex analytical conditions are transformed to linear ones. For each performance index, sufficient conditions are given in terms of LMIs. (4) The FD scheme is proposed based on the obtained conditions which is valid to detect the fault in finite-frequency domain for systems with large values of Lipschitz constants. Also, it has a better performance in saving the communication resources and detecting faults than the existing one. The rest of this paper is organized as follows: Section 2 gives the system description and the problem formulation. In Section 3, the LPV-based approach for Lipschitz systems is proposed. LMI-based fault detection filter design conditions under event-triggered scheme are then derived in the finitefrequency domain in Section 4. Two examples are given in Section 5 to show the advantages of the proposed method. Section 6 concludes the paper. Notations: In this paper, the notations are standard. For a matrix A, AT , A⊥ denote its transpose and orthogonal complement, respectively and A > 0(A < 0) means that it is positive definite (negative definite). The symbol ∗ will be used in some matrix expressions to denote the transposed elements in the symmetric positions of a matrix. In is an identity matrix of size n. The sum of a square matrix A and its transpose AT is denoted by ith z}|{ T He(A) := A + A . es (i) = (0, · · · , 0, 1 , 0, · · · , 0)T ∈ Rs , s ≥ 1 is a vector {z } | s components

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of the canonical basis of Rs .

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2. System description and problem statement

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2.1. System description Consider the following nonlinear networked control systems described by  x(k + 1) = Ax(k) + Φ(x(k)) + Bf f (k) + Bd d(k) (1) y(k) = Cx(k) + Df f (k) + Dd d(k)

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where x(k) ∈ Rn is the state vector, y(k) ∈ Rp is the output vector, d(k) ∈ Rl is the unknown bounded disturbance vector, f (k) ∈ Rq is the fault to be detected. Matrices A ∈ Rn×n , Bf ∈ Rn×q , Bd ∈ Rn×l , C ∈ Rp×n , Df ∈ Rp×q , Dd ∈ Rp×l are real known constant matrices. The nonlinear function Φ(x(k)) is assumed to satisfy Φ(0) = 0 and rΦ -Lipschitz, i.e.: kΦ(x(k)) − Φ(ˆ x(k))k ≤ rΦ kx(k) − xˆ(k)k, ∀x(k), xˆ(k) ∈ Rn

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where rΦ is a known Lipschitz constant.

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Remark 1. Note that globally Lipschitz nonlinear systems are only a limited class of nonlinear systems. When Φ(x(k)) is locally Lipschitz, all the results derived in this paper are valid in a neighborhood around a nominal point. Let the event-triggered FD filter be of the following form:  x(k) + Φ(ˆ x(k)) + L(˜ y (k) − yˆ(k))  xˆ(k + 1) = Aˆ yˆ(k) = C xˆ(k) (3)  r(k) = V (˜ y (k) − yˆ(k))

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where xˆ(k) ∈ Rn and yˆ(k) ∈ Rp represent the state and output estimation vectors, respectively. r(k) ∈ Rs is the residual signal. y˜(k) ∈ Rp is the last released data from the sensor to the filter modul. L and V are the filter matrices to be determined. For the purpose of saving the limited communication resources, the eventtriggered scheme in [8] is introduced to decide whether or not the sampled data should be transmitted. The event instants are assumed to be k0 , k1 , k2 , · · · , where k0 = 0 is the initial event-triggered time. Define y˜(k) = y(ki ), k ∈ [ki , ki+1 ).

The event-triggered FD scheme is shown in Fig. 1. 5

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Figure 1: The schematic of event-triggered FD problem.

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In order to reduce the data transmission frequency, the current measurement y(k) satisfying δ T (k)Ωδ(k) ≥ σy T (k)Ωy(k)

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will be released, where δ(k) = y˜(k) − y(k) is the difference between the last released data of the generator and the current output of the plant, σ ∈ [0, 1) is a scalar, and Ω ∈ Rm×m is a positive weighting matrix.

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Remark 2. When σ = 0, {k0 , k1 , k2 , · · · } = {0, 1, 2, · · · }, all the output signals will be transmitted to the FD module. In this case, the event-triggered scheme reduces to the case of time-triggered scheme. Let

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e(k) = x(k) − xˆ(k), η(k) = [xT (k) eT (k)]T , ∆Φ(k) = Φ(x(k)) − Φ(ˆ x(k)). (6)

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Considering (1), (3) and the event-triggered scheme (5), the filter error system can be derived as follows:   η(k + 1) = Aη(k) + I1 Φ(x(k)) + I2 ∆Φ(k) + Bf f (k) + Bd d(k) + Bδ δ(k) r(k) = C1 η(k) + Df f (k) + Dd d(k) + Dδ δ(k)  y(k) = C2 η(k) + Df f (k) + Dd d(k) (7)

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where

        A 0 I 0 Bf A= , I1 = , I2 = , Bf = , 0 A − LC 0 I Bf − LDf         Bd 0 Bd = , Bδ = , C1 = 0 V C , C2 = C 0 , Bd − LDd −L

Df = V Df , Dd = V Dd , Dδ = V. 6

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Ωθ := {θ ∈ R|ϑ1 ≤ θ ≤ ϑ2 }

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2.2. Problem statement In this subsection, the finite-frequency H∞ performance index and finitefrequency H− index are defined to formulate the fault detection problem. Consider the following finite frequency interval for frequency θ of the corresponding signal:

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where 0 < ϑ2 − ϑ1 ≤ 2π, ϑ1 , ϑ2 are given real scalars and used to describe the middle-frequency range of the corresponding signal. Note that, when −ϑ1 = ϑ2 = ϑl , Ωθ denotes the low frequency range

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Ωθ := {θ ∈ R||θ| ≤ ϑl }

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where ϑl is assumed to be a known positive real scalar; when ϑ1 = ϑh , ϑ2 = 2π − ϑh , Ωθ denotes the high frequency range Ωθ := {θ ∈ R||θ| ≥ ϑh }

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where ϑh is assumed to be a known positive real scalar.

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Definition 1. The filtering error system (7) has a finite-frequency H∞ index bound γ, if under zero initial condition, the following inequality (11)

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kr(k)k2 ≤ γkd(k)k2

holds for all solutions of (7) with d(k) ∈ l2 in finite-frequency range ϑd1 ≤ θd ≤ ϑd2 , such that ∞ X

(η(k + 1) − ejϑd1 η(k))(η(k + 1) − e−jϑd2 η(k))T ≤ 0

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jϑd$

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where θd denotes the frequency of d(k), ϑd$ = (ϑd2 − ϑd1 )/2, ϑd1 and ϑd2 reflect the frequency range of d(k). γ is a given real positive scalar which denotes the worst case criterion for the effect of d(k) on the residual r(k). Then the parameter γ is called the finite-frequency H∞ index bound of the control. The smaller the γ is, the smaller the influence of disturbance d(k) on the residual r(k). Definition 2. The filtering error system (7) has a finite-frequency H− index bound β, if under zero initial condition, the following inequality kr(k)k2 ≥ βkf (k)k2 7

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holds for all solutions of (7) with f (k) in finite-frequency range ϑf 1 ≤ θf ≤ ϑf 2 such that ejϑf $

∞ X (η(k + 1) − ejϑf 1 η(k))(η(k + 1) − e−jϑf 2 η(k))T ≤ 0

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k=0

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where θf denotes the frequency of f (k), ϑf $ = (ϑf 2 − ϑf 1 )/2, ϑf 1 and ϑf 2 reflect the frequency range of f (k). The larger the β is, the greater the influence of fault f (k) on the residual r(k).

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Remark 3. Definitions 1 and 2 are motivated by the work of [37] and [39], which are given from a time-domain perspective and can be regarded as extensions of the standard H∞ and H− performances. If ϑl = π, ϑ1 = −π, ϑ2 = π, or ϑh = 0, the constraints (12) and (14) are automatically satisfied and the finite-frequency range reduces to full frequency range.

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Remark 4. Note that the GKYP lemma ([33]) based FD methods are developed from the viewpoint of transfer function matrix which makes them cannot be directly applied for Lipschitz nonlinear systems. However, [39] has provided time domain interpretations of finite-frequency properties. As stated in the literature, the time domain characterization of these properties are useful for capturing finite-frequency specifications for nonlinear systems because it is given in terms of the input/output signals rather than the transfer function. Recently, some results in finite frequency domain for nonlinear systems have been reported, such as [37, 38] for fuzzy systems whose subsystems are linear time-invariant system. However, it is hard to use these existing finite-frequency FD results for this class of Lipschitz nonlinear networked systems subject to the event-triggered scheme. The FD problem considered in this paper can be formulated as follows: For a given Lipschitz nonlinear system (1) and two prescribed levels of disturbance attenuation and fault sensitivity γ > 0 and β > 0, design a nonlinear filter in the form of (3) such that under the event-triggered condition (5), the following three specifications are satisfied: (S.1) The filtering error system (7) is asymptotically stable. (S.2) To reduce the effects of disturbances to the residual, the filtering error system (7) satisfies inequality kr(k)k2 ≤ γkd(k)k2

under constraint (12). (S.3) To increase the effects of faults to the residual, the filtering error system 8

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(7) satisfies inequality

with f (k) ∈ l2 under constraint (14). 3. LPV-based approach for Lipschitz systems

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kr(k)k2 ≥ βkf (k)k2

The following lemma will be useful in the later developments.

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Lemma 1.[21] Considering the function Φ : Rn → Rn , the two following items are equivalent: (a) Lipschitz property: Φ is rΦ -Lipschitz with respect to its argument, i.e.: kΦ(x) − Φ(y)k ≤ rΦ kx − yk,

∀x, y ∈ Rn .

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(b) Reformulated Lipschitz property: for all i, j = 1, · · · , n, there exist functions Φij : Rn × Rn → R and constants rΦij and r¯Φij , so that ∀x, y ∈ Rn ,

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Φij Hij (x − y)

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and rΦij ≤ Φij ≤ r¯Φij , where Φij = Φij (xyj−1 , xyj ) and Hij = en (i)eTn (j).

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Remark 5. More details about the functions Φij in (16) can be found in [21]. As stated in [21], the standard LMI methods are generally infeasible for systems with large Lipschitz constants. Lemma 1 takes the detailed structure of the nonlinearity into account which can reduce the conservatism of the existing synthesis conditions for Lipschitz systems. According to Lemma 1, the Lipschitz property on Φ(x(k)) leads to the existence of bound functions Φij : Rn × Rn → R and constants rΦij and r¯Φij , for all i, j = 1, · · · , n, so that rΦij ≤ Φij ≤ r¯Φij P P ∆Φ(k) = [ ni=1 nj=1 Φij (xxˆj−1 , xxˆj )Hij ]e(k)

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where Hij = en (i)eTn (j). To simplify the presentation, let us introduce the following notations: Λ :=

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Φij (x0j−1 , x0j )Hij , Υ :=

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Φij (xxˆj−1 , xxˆj )Hij

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and A(Λ) = A + Λ, A(Υ) = A + Υ.

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Therefore, the filtering error system (7) can be re-organized into a compact form   η(k + 1) = A(Λ, Υ)η(k) + Bf f (k) + Bd d(k) + Bδ δ(k) r(k) = C1 η(k) + Df f (k) + Dd d(k) + Dδ δ(k) (21)  y(k) = C2 η(k) + Df f (k) + Dd d(k) where

  A(Λ) 0 A(Λ, Υ) = . 0 A(Υ) − LC

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Notice that in view of Lemma 1, the parameters Λ and Υ belong to bounded convex set Hn for which the set of vertices is defined by VHn = {Γ ∈ Rn×n , Γij ∈ {rΦij , r¯Φij }}.

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˜ Υ) ˜ the vertices in VHn × VHn corresponding In the sequel, we denote by (Λ, to the bounded convex set Hn × Hn .

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Remark 6. In this section, the nonlinear filter error system (7) is rewritten under a LPV form by the use of a reformulated Lipschitz property. Then the considered FD problem is transformed to the problem of H− /H∞ filter design for a class of LPV systems (21) under event-trigged scheme in finitefrequency domain.

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4. Main results

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In this section, the finite frequency H− /H∞ filter design conditions will be given. Before proceeding further, a new lemma is derived which provides sufficient conditions for the desired performances.

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Lemma 2. Assume that the filtering error system (21) is asymptotically stable. Letting β > 0 be a given constant, if there exist symmetric matrices P1 , Q1 > 0 and Ω > 0 such that the following inequality holds, ∀(Λ, Υ) ∈ Hn × H n ,  T   A(Λ, Υ) Bf Bδ A(Λ, Υ) Bf Bδ Ξf (23) I 0 0 I 0 0  T    T   C1 Df Dδ C1 Df Dδ C2 Df 0 C2 Df 0 + Πf + ‫ג‬ <0 0 I 0 0 I 0 0 0 I 0 0 I 10

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   −I 0 σΩ 0 where Πf = ,‫=ג‬ and 0 β 2I 0 −Ω 1) For the low-frequency range |θf | ≤ ϑf l ,   P1 Q1 Ξf = Q1 −P1 − 2cos(ϑf l )Q1

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2) For the middle-frequency range ϑf 1 ≤ θf ≤ ϑf 2 ,   P1 ejϑf c Q1 Ξf = −jϑf c e Q1 −P1 − 2cos(ϑf $ )Q1

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with ϑf c = (ϑf 2 + ϑf 1 )/2, ϑf $ = (ϑf 2 − ϑf 1 )/2. 3) For the high-frequency range |θf | ≥ ϑf h ,   P1 −Q1 Ξf = −Q1 −P1 + 2cos(ϑf h )Q1

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then system (21) satisfies the finite-frequency H− performance kr(k)k2 ≥ βkf (k)k2 .

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Proof. We first consider the middle-frequency case for the system (21) with d(k) = 0. Inspired by [39], multiplying the inequality (23) by [η T (k) f T (k) δ T (k)] from the left and by its transpose from the right, and recalling the fact uT Q1 v = tr(Q1 vuT ) for arbitrary vector u and v, one can derive − η T (k)P1 η(k) + η T (k + 1)P1 η(k + 1) − rT (k)r(k) + β 2 f T (k)f (k) jϑf c

+ tr(Q1 (e

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−jϑf c

η(k)η (k + 1) + e

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T

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η(k + 1)η (k) − 2cos(ϑf $ )η(k)η (k)))

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− δ (k)Ωδ(k) + σy (k)Ωy(k) ≤ 0.

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Considering the event-triggered communication scheme (5), for every k ∈ [ki , ki+1 ), δ T (k)Ωδ(k) < σy T (k)Ωy(k), then we have − η T (k)P1 η(k) + η T (k + 1)P1 η(k + 1) − rT (k)r(k) + β 2 f T (k)f (k)

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+ tr(Q1 (ejϑf c η(k)η T (k + 1) + e−jϑf c η(k + 1)η T (k) − 2cos(ϑf $ )η(k)η T (k))) ≤ 0.

Since η(0) = 0 and the system (21) is stable, taking the summation from k = 0 to ∞, we have ∞ X k=0

(−rT (k)r(k) + β 2 f T (k)f (k)) + tr(Q1 M ) ≤ 0 11

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where ∞ X k=0

(ejϑf c η(k)η T (k + 1) + e−jϑf c η(k + 1)η T (k) − 2cos(ϑf $ )η(k)η T (k))

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M :=

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According to Euler’s formula, it can be verified that −M is equal to the lefthand side of (14) after some calculations. So the M is positive semidefinite. Since Q1 > 0, the last term on the left-hand side ofP (29) is nonnegative when P∞ T 2 T (14) is satisfied. Hence we have k=0 r (k)r(k) ≥ ∞ k=0 β f (k)f (k), which is equivalent to condition (13) for middle-frequency domain in Definition 2. Similarly, the results follow by choosing ϑf 2 := ϑf l and ϑf 1 = −ϑf l for the low-frequency case and ϑf 2 := 2π − ϑf h and ϑf 1 = ϑf h for the high-frequency case. Thus, the proof is completed. Remark 7. The conditions to capture the fault sensitivity performance subject to event-triggered scheme in finite-frequency domain are given in Lemma 2. However, it contains coupled matrix variables, which leads to a non-convex problem. In the following, decoupling techniques will be developed to transform the analytical conditions to linear ones.

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Lemma 3.(Projection Lemma)[40] Given a symmetric matrix Ψ and two matrices U, V, the problem

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Ψ + UXV T + VX T U T < 0

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is solvable with respect to decision matrix X if and only if T

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U ⊥ ΨU ⊥ < 0,

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V ⊥ ΨV ⊥ < 0

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where U ⊥ and V ⊥ denote the orthogonal complement of U and V, respectively.

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Lemma 4.[41] Let R1 = R1T > 0 and R2 matrices of appropriate size. The following expression holds: −R2T R1−1 R2 ≤ R1 − R2 − R2T .

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4.1. Fault sensitivity condition Theorem 1. Assume the filtering error system (21) is asymptotically stable. For two given constants β > 0, ε1 > 0 and a given matrix S with compatible dimension, if there exist matrices Ω > 0, V¯ , W11 , W12 , W , det(W ) 6= 0, Y 12

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P11 and symmetric matrices P1 = ∗ ˜ Υ) ˜ ∈ VHn × VHn , the following ∀(Λ,  Γ11 Γ12 Γ13 Γ14  ∗ Γ22 Γ23 Γ24   ∗ ∗ Γ33 Γ34   ∗ ∗ ∗ Γ44   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ where

   P12 Q11 Q12 , Q1 = > 0 such that P13 ∗ Q13 inequality holds:  Γ15 −ε1 Y  Γ25 −Y   Γ35 0 <0 (33) T ¯  Γ45 −C V  Γ55 −DfT V¯ − S T Y  ∗ −Ω − V¯

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Γ23 = Γ25 = Γ34 =

11 T W11 Bf − W12 S + ε1 W T Bf − ε1 Y Df , Γ22 = P13 − W − W T , T ˜ Γ24 = µ1 Q13 + W T A(Υ) ˜ − Y C, µ1 QT12 + W12 A(Λ), T −W S + W12 Bf + W T Bf − Y Df , Γ33 = −P11 − µ2 Q11 + σC T ΩC, ˜ T W12 S, −P12 − µ2 Q12 , Γ35 = σC T ΩDf + A(Λ)

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Γ15 =

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T Γ11 = P11 − W11 − W11 , Γ12 = P12 − W12 − ε1 W T , ˜ Γ14 = µ1 Q12 + ε1 W T A(Υ) ˜ − ε1 Y C, Γ13 = µ1 Q11 + W T A(Λ),

˜ T W S − C T Y T S, Γ44 = −P13 − µ2 Q13 − C T V¯ C, Γ45 = −C T V¯ Df + A(Υ) Γ55 = β 2 I − DT V¯ Df + σDT ΩDf + He(B T W S − DT Y T S + B T W12 S) f

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and Y = W L, V¯ = V V , µ1 = 1, µ2 = 2cos(ϑf l ) for the fault in lowfrequency domain |θf | ≤ ϑf l ; µ1 = ejϑf c , µ2 = 2cos(ϑf $ ) for the fault in middle-frequency domain ϑf 1 ≤ θf ≤ ϑf 2 ; µ1 = −1, µ2 = −2cos(ϑf h ) for the fault in high-frequency domain |θf | ≥ ϑf h , then the system (21) satisfies the finite-frequency H− performance index kr(k)k2 ≥ βkf (k)k2 . T

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Proof. Based on Lemma 2, the system (21) has a finite frequency H− performance index β if (23) holds. We first consider the low-frequency case for the system (21) with d(k) = 0. By matrix manipulations, the inequality (23) can be rewritten as    T A(Λ, Υ) Bf Bδ A(Λ, Υ) Bf Bδ   I 0 0 0 0  Ψf  I  < 0, ∀(Λ, Υ) ∈ Hn × Hn   0  0 I 0 I 0 0 0 I 0 0 I (34) 13

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  P1 Q1 0 0  ∗ Ψf 22 −C1T Df + σC2T ΩDf −C1T Dδ   Ψf =  T 2 T ∗ −DfT Dδ  ∗ β I − Df Df + σDf ΩDf ∗ ∗ ∗ −Ω − DδT Dδ

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where

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with Ψf 22 = −P1 − 2cos(ϑf l )Q1 − C1T C1 + σC2T ΩC2 . The following null space bases calculations yield  ⊥   T −I A (Λ, Υ) I 0 0 T A (Λ, Υ)   =  BfT 0 I 0  BfT  T Bδ 0 0 I BδT

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From Lemma 3, the following inequality is a sufficient condition for (34) ∀(Λ, Υ) ∈ Hn × Hn ,

Ψf + Uf Xf VfT + Vf XfT UfT < 0

which has the same form of inequality (30) with  T Uf = −I A(Λ, Υ) Bf Bδ , Vf = I.

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<0

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The matrix Xf is an introduced additional variable and can be chosen as the   form of Xf = W1 Sf , where Sf = I 0 S1 0 . Consequently, the inequality (36) can be rewritten into   W1T Bδ P1 − He(W1 ) Q1 + W1T A(Λ, Υ) −W1 S1 + W1T Bf   ∗ ϕ22 ϕ23 −C1T Dδ   T T T  ∗ ∗ ϕ33 −Df Dδ + S1 W1 Bδ  ∗ ∗ ∗ −Ω − DδT Dδ (38)

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where ϕ22 = −P1 − 2cos(ϑf l )Q1 − C1T C1 + σC2T ΩC2 , ϕ23 = −C1T Df + σC2T ΩDf AT (Λ, Υ)W1 S1 , ϕ33 = β 2 I − DfT Df + σDfT ΩDf + He(BfT W1 S1 ).

+

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Therefore, by using the convexity principle discussed in [42], we deduce ˜ Υ) ˜ ∈ VHn × that the inequality (38) holds if it can be verified for all (Λ, V the matrices W1 and S1 therein into the form of W1 =  Hn . Partitioning    W11 W12 0 and S1 = , where W is a nonsingular matrix and denoting ε1 W W S Y = W T L, V¯ = V T V , after some matrix manipulation, the condition (33) is obtained for fault in low-frequency domain. Further, by using similar techniques, we can obtain the sufficient conditions for middle- and high-frequency cases. The proof is completed. 14

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4.2. Disturbance attenuation condition

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Theorem 2. Assume the filtering error system (21) is asymptotically stable. For two given constants γ > 0 and ε2 , if there exist matricesW21 , W22, Ω > 0, P21 P22 V¯ , W , det(W ) 6= 0, Y and symmetric matrices P2 = , Q2 = ∗ P23   Q21 Q22 ˜ Υ) ˜ ∈ VHn × VHn , the following inequality > 0 such that ∀(Λ, ∗ Q23 holds:   Θ11 Θ12 Θ13 Θ14 Θ15 −ε2 Y  ∗ Θ22 Θ23 Θ24 Θ25 −Y    T  ∗  ∗ Θ Θ σC ΩD 0 33 34 d   (39) T ¯ T ¯  < 0  ∗ ∗ ∗ Θ44 C V Dd C V    ∗ ∗ ∗ ∗ Θ55 DdT V¯  ∗ ∗ ∗ ∗ ∗ −Ω + V¯ where

Θ24 = Θ33 =

21 T ˜ P23 − W − W , Θ23 = + W22 A(Λ), T ˜ − Y C, Θ25 = W22 µ3 Q23 + W T A(Υ) Bd + W T Bd −P21 − µ4 Q21 + σC T ΩC, Θ34 = −P22 − µ4 Q22 , T

µ3 QT22

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Θ22 =

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T T ˜ Θ11 = P21 − W21 − W21 , Θ12 = P22 − W22 − ε2 W T , Θ13 = µ3 Q21 + W21 A(Λ), ˜ − ε2 Y C, Θ15 = W T Bd + ε2 W T Bd − ε2 Y Dd , Θ14 = µ3 Q22 + ε2 W T A(Υ)

− Y Dd ,

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Θ44 = −P23 − µ4 Q23 + C T V¯ C, Θ55 = −γ 2 I + DdT V¯ Dd + σDdT ΩDd

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and Y = W T L, V¯ = V T V , µ3 = 1, µ4 = 2cos(ϑdl ) for the disturbance in lowfrequency domain |θd | ≤ ϑdl ; µ3 = ejϑdc , µ4 = 2cos(ϑd$ ) for the disturbance in middle-frequency domain ϑd1 ≤ θd ≤ ϑd2 ; µ3 = −1, µ4 = −2cos(ϑdh ) for the disturbance in high-frequency domain |θd | ≥ ϑdh , then the system (21) satisfies the finite-frequency H∞ performance index kr(k)k2 ≤ γkd(k)k2 .

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Proof. We first consider the high-frequency case for the system (21) with f (k) = 0. Following the process of the proof of Theorem 1, if there exist symmetric matrices P2 , Q2 > 0, Ω > 0 such that the following inequality

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holds:  T   A(Λ, Υ) Bd Bδ A(Λ, Υ) Bd Bδ  I  0 0 0 0   Ψd  I  < 0, ∀(Λ, Υ) ∈ Hn × Hn ,  0  0 I 0 I 0 0 0 I 0 0 I (40) where

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  P2 −Q2 0 0   ∗ Ψd22 C1T Dd + σC2T ΩDd C1T Dδ  Ψd =  T T 2 T ∗ Dd Dδ  ∗ −γ I + Dd Dd + σDd ΩDd ∗ ∗ ∗ −Ω + DδT Dδ

with Ψd22 = −P2 + 2cos(ϑdh )Q2 + C1T C1 + σC2T ΩC2 , then the system (21) satisfies the finite-frequency H∞ performance index kr(k)k2 ≤ γkd(k)k2 . Applying Lemma 2, we can obtain that the following inequality is a sufficient condition for (40), ∀(Λ, Υ) ∈ Hn × Hn ,

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Ψd + Ud Xd VdT + Vd XdT UdT < 0.

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where

(41)

 T Ud = −I A(Λ, Υ) Bd Bδ , Xd = W2 S2 , Vd = I.

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Therefore, by using the convexity principle discussed in [42], we deduce that ˜ Υ) ˜ ∈ VHn × VHn . Parthe inequality (41) holds if it can be verified for all (Λ,   W21 W22 titioning the matrices W2 and S2 therein into the form of W2 = ε2 W W   and S2 = I 0 0 0 , where W is a nonsingular matrix. Denoting Y = W T L and V¯ = V T V , after some matrix manipulation, the condition (39) is obtained for disturbance in high-frequency domain. Further, by using similar techniques, we can obtain the sufficient conditions for low- and middle-frequency cases. The proof is completed. Remark 8. The considered FD problem is actually a multi-constrained design problem which may have no solution due to the increased number of constraints. Introducing different matrices P1 , P2 and Q1 , Q2 for each constraint can bring more degrees of freedom. However, the coupling of the 16

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matrix variables in (34) and (40) results in a non-convex problem, so (34) and (40) cannot be handled by the linear optimization procedures. To recover convexity, it is required that P1 = P2 > 0 and Q1 = Q2 = 0 in full frequency domain. In finite-frequency domain, the Projection Lemma is exploited to separate L from P1 , P2 , Q1 and Q2 . Then novel sufficient conditions are obtained in (33) and (39) and matrices P1 , P2 , Q1 and Q2 can be chosen to be different from one another.

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4.3. Stability condition Then we establish the stability of the filter error system. Note that the filter design conditions given in the previous subsections cannot ensure a stable system, so an additional constraint should be added to guarantee the stability of the system (21).

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Theorem 3. For a given constant ε3 , ifthere exist matrices Ω > 0, G11 , G12 ,  P31 P32 ˜ Υ) ˜ ∈ VHn × VHn , > 0 such that ∀(Λ, W , det(W ) 6= 0, Y and P3 = ∗ P33 the following inequality holds:   k11 k12 k13 k14 −Y  ∗ k22 k23 k24 −ε3 Y     ∗ <0 ∗ k −P 0 (42) 33 32    ∗ ∗ ∗ −P33 0  ∗ ∗ ∗ ∗ −Ω

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where

˜ k11 = P31 − G11 − GT11 , k12 = P32 − W T − GT12 , k13 = G11 A(Λ), ˜ − Y C, k22 = P33 − ε3 W − 3 W T , k23 = G12 A(Λ), ˜ k14 = W T A(Υ)

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˜ − 3 Y C, k33 = −P31 + σC T ΩC, k24 = ε3 W T A(Υ)

and Y = W T L, then the system (21) is asymptotically stable.

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Proof. Consider the following Lyapunov functional candidate for system (21) V (k) = η(k)T P3 η(k)

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with P3 = P3T > 0. According to δ T (k)Ωδ(k) < σy T (k)Ωy(k), we have ∆V (k) = V (k + 1) − V (k)

≤ η T (k)[AT (Λ, Υ)P3 A(Λ, Υ) − P3 ]η(k) + δ T (k)BδT P3 Bδ δ(k)

+ ση T (k)C2T ΩC2 η(k) − δ(k)T Ωδ(k) = χ(k)T Ψs χ(k)

where

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  T  η(k) A (Λ, Υ)P3 A(Λ, Υ) − P3 + σC2T ΩC2 AT (Λ, Υ)P3 Bδ , Ψs = χ(k) = . δ(k) ∗ BδT P3 Bδ − Ω (45)

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+ δ T (k)BδT P3 A(Λ, Υ)η(k) + η T (k)AT (Λ, Υ)P3 Bδ δ(k)

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If Ψs < 0 holds, then system (21) is asymptotically stable. By the Schur complement, the problem is concluded to satisfy   −P3 P3 A(Λ, Υ) P3 Bδ  ∗ −P3 + σC2T ΩC2 0  < 0, (46) ∗ ∗ −Ω

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then pre and post-multiplying by diag (GP3−1 , I2n , I2n ) and its transpose, based on Lemma 4, we can obtain that the following inequality is a sufficient condition for (46)   P3 − G − GT GA(Λ, Υ) GBδ  0 <0 ∗ −P3 + σC2T ΩC2 (47) ∗ ∗ −Ω

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where G is a slack matrix. Therefore, by using the convexity principle we ˜ Υ) ˜ ∈ VHn × deduce that the inequality (47) holds if it can be verified for all (Λ,  G11 W T VHn . Partitioning the matrix G therein into the form of G = , G12 ε3 W T where W is a nonsingular matrix and denoting Y = W T L, after some matrix manipulation, the condition (42) is obtained. The proof is completed. Remark 9. Although the LPV method can provide solutions even for large Lipschitz constants, the drawback is that more demanding LMIs needed to be solved. In [21], all the LMIs are enforced by a common Lyapunov matrix 18

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which generally provides quite conservative results or no solution due to the increased number of LMIs. By slack variable technique, the LMI feasibility conditions are derived in which the product of matrix variable P3 and the filter gain matrix L is not involved. Furthermore, this extra matrix W does not present any structural constraints, so the conservatism is reduced compared with the one in [21]. 4.4. Algorithm By combining Theorems 1, 2 and 3, specifications 1) − 3) given in Section 2 will be satisfied if LMIs (33), (39) and (42) hold simultaneously. The following theorem is given to solve the FD problem.

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Theorem 4. Let two prescribed performance indices γ, β, three constants ε1 , ε2 , ε3 and a matrix S be given, system (21) subject to the event-triggered scheme (5) is asymptotically stable and satisfies conditions (11) and (13), if there exist matrices  Ω > 0, W11 , W12 , W21 , W  22 , G11 ,G12 , W , det(W ) 6= 0, P11 P12 P21 P22 P31 P32 Y , V¯ and P1 = , P2 = , P3 = > 0, Q1 = ∗ P ∗ P ∗ P33 13 23     Q11 Q12 Q21 Q22 > 0, Q2 = > 0 such that (33), (39) and (42) hold. ∗ Q13 ∗ Q23 T Furthermore, the FD filter matrices L = W −1 Y , and V can be obtained by means of the matrix V¯ , where V is a factorization of V¯ (i.e., V¯ = V T V ). Finally, the feasible solutions to the FD problem are solved via the following optimization problem: max β s.t.(33), (39), (42)

(48)

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for the given scalar γ. Now, for designing the fault detection filter in finite-frequency domain, one can follow the following design procedures: step 1: Compute Φij , rΦij , r¯Φij , i, j = 1, · · · , n. step 2: Give the predefined parameter σ and γ. Construct LMIs (33), (39), (42) and solve (48) to obtain W , Y and V¯ . If there is a solution, then it means T that the FD filter exists. Subsequently, we can compute L by L = W −1 Y and V by V¯ = V T V . Otherwise, the FD filer dose not exist. Consequently, the fault detection filter can be constructed as in (3). Remark 10. It should be claimed that, in the interest of a better fault sensitivity performance, we solve the optimization problem (48) under an 19

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acceptable attenuation level γ rather than the best. To balance these two performances, we can solve the following optimization problem: (49)

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min λ1 γ − λ2 β s.t.(33), (39), (42)

where λ1 , λ2 > 0 are weighting factors subject to λ1 + λ2 = 1. As pointed out in [23], a satisfactory FD performance can only be achieved by a proper trade-off between the robustness and sensitivity. A smaller γ will result in a smaller β, that is, a better disturbance attenuation performance can be obtained by sacrificing some level of fault sensitivity performance.

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Remark 11. When solving the conditions of Theorem 4, ε1 , ε2 , ε3 and S need to be given in advance such that (48) becomes a convex optimization problem. The role of the weighting matrix Ω is to enhance the feasibility of the optimization problem (48).

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Remark 12. It should be pointed out that considerable computational burdens may arise when the complexity of the function Φ(x(k)) increases. The cost of more demanding LMIs does not play an important role on the feasibility of (48) because the LPV method can provide less restrictive LMI synthesis conditions. Besides, the constant filter gains are computed off line for the proposed method, so the time taken to solve the LMIs will have no effect on the real-time applications.

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4.5. Detection threshold design In this section, the threshold for detecting faults is designed and the detection logic unit is based on the results proposed by [43]. The residual evaluation function Jr (τ ) is chosen as v u kτ u1 X rT (k)r(k) (50) Jr (τ ) = t τ k=k 1

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where k1 denotes the initial evaluation time instant and kτ stands for the evaluation time. The threshold Jth is determined by Jth =

sup

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f =0,d(k)∈L2

and the faults can be detected using the following logical relationship: Jr > Jth ⇒ fault alarm, Jr ≤ Jth ⇒ no fault. 20

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5. Examples

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Two examples are given to illustrate the effectiveness and applicability of the proposed method. 0.2 0.15 0.1 0.05 0

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Figure 2: Disturbance.

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5.1. Example 1 Consider system (1) with parameters as follows:         −0.1 0.2 0.5 0.2 A= , Bf = , Bd = , Dd = 0.5, C = −1 1 . −0.3 0.1 0.1 −0.2

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It is assumed that the external disturbance signal is in high-frequency domain ϑdh = π/1.2, which is shown in Figure 2, and the fault is in low-frequency domain ϑf l = π/15. Given γ = 0.8, we solve the optimisation problem (48) in the following simulation cases: Case 1 (strictly proper case):    0.1sin(x2 (k)) 0.4, 10 ≤ k ≤ 80 Df = 0, f (k) = , Φ(x(k)) = . 0, otherwise 0.1sin(x1 (k)) According to Lemma 1, we have VHn =



 0 ±0.1 . ±0.1 0 21

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Different values of σ are considered and the corresponding performance indices β are given in Table 1. It can be seen from Table 1 that the larger the σ, the worse the fault sensitivity performance. On investigating the Table 1: Comparison of the H− performance β for different σ

σ σ=0.05 σ=0.1 σ=0.2 σ=0.3 σ=0.4 β 0.7768 0.5938 0.4018 0.2792 0.1786

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performance of the proposed FD filter, the initial conditions are selected as  T  T x(0) = 0.01 −0.01 , xˆ(0) = 0 0 . When σ = 0.1, the residual response and residual evaluation response are displayed in Fig. 3. Fig. 4 demonstrates the release instants and release intervals of the event detector. It is shown that the fault can be detected timely and the proposed method is effective for strictly proper system. In the simulation time 10s, we observer that only 69 times are triggered. This result compares favorably with the time-triggered communication scheme (100 times). For further comparison, we also simulate the residual response and residual evaluation function response in the context of the event-triggered scheme parameter σ = 0.4, which are shown in Fig. 3. The release instants and release intervals of the event detector are shown in Fig. 4. In the simulation time, 46 times are triggered which is less than that in the context of σ = 0.1. The fault can also be detected, 22

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Figure 4: Case 1: Release interval with σ = 0.1 (left) and release interval with σ = 0.4 (right).

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but residual evaluation function response is not as sensitive as the one with σ = 0.1 because of the less transmission of data. To illustrate the advantage of the proposed approach, we compare it with the existing full-frequency method ([16, 17]) in the following case since the method in [16, 17] is not applicable for strictly proper systems, i.e. Case 1. Case 2 (non-proper case):  iT h r 0.2, 20 ≤ k ≤ 90 (x1 (k) + |x1 (k)|) , f (k) = Df = 1, Φ(x(k)) = 0 0, otherwise. 2

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we have

VHn =



   0 0 0 0 , 0 0 r 0

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where r > 0 is used to boost the comparison. The Lipschitz constant of the nonlinearity is rΦ = r. When σ = 0.1, the proposed method can tolerate Lipschitz constants of not more than 1.78 while the existing one can only tolerate Lipschitz constants of not more than 0.48. Choosing r = 0.4, ε1 = 0,  T ε2 = 0, ε3 = 1 and S = −0.36 −0.56 , we can obtain the optimal value by Theorem 4 for sensitivity performance is β = 0.8962 with the filter gains  T L = 0.2188 −0.7911 , V = 0.7075. The optimal value obtained by the existing full frequency method in [16, 17] for sensitivity performance is β = 0.6121 with the filter gains  T L = 0.4608 0.0496 , V = 0.8582. 23

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The residual response and residual evaluation response by these two methods 1.4 by our method by existing method

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Figure 5: Case 2: Generated residual |r(k)| (left) and residual evaluation function Jr (k) (right).

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are displayed in Fig. 5. From which, it is easy to see that fault alarms are delivered timely and both of these two methods are effective for non-proper systems, i.e. Df 6= 0. However, the generated residual in this paper is more sensitive to fault than the one in full frequency domain. Furthermore, only 76 times are triggered by our method which can be seen in Fig. 6. These results indicate that the designed FD filter in finite-frequency domain has a better performance in saving the communication resources and detecting fault than the existing one.

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5.2. Example 2 Consider the model of an inverted pendulum controlled by a dc motor via a gear train borrowed from [44], where the states are the angular position, velocity of the inverted pendulum and the current of the dc motor xT = (x1 , x2 , x3 ) = (θp , ωp , Ia ), while the output is θp . The state-space model is  θ˙p = ωp     N Km g Ia ω˙ p = sin(θp ) + l ml2   KN R 1   I˙a = − b ωp − a Ia + u La La La

where Km is the motor torque constant, Kb is the back emf constant, and N is the gear ratio. The parameters for the plant are g = 9.8 m/s2 , l = 1m, 24

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5 4.5 4

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m = 1kg, N = 10, Km = 0.1 Nm/A, Kb = 0.1 Vs/rad, Ra = 1Ω, La = 100 mH. These parameters lead to          0 0 0 1 0 x1 x˙ 1 x˙ 2  = 0 0 1  x2  +  0  u + 9.8sin(x1 ) 10 0 0 −10 −10 x3 x˙ 3     x1 y = 1 0 0 x2  x3

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Using the Euler’s discretization method with a sampling period T = 0.1s, a discrete-time model of the system is obtained. The control input of this simu- lation is chosen as u(k) = Kx(k), where K = −19.3239 −7.7777 −0.6908 . Considering that there exist the disturbance input and the fault, the closedloop state-space description is as (1) with system parameters as follows:       1 0.1 0 0 0      0 1 0.1 A= , Bf = −0.01 , Bd = 0.2 , −19.3239 −8.7777 −0.6908 0.02 0.1 

 0   C = 1 0 0 , Df = 1, Dd = 0.1, Φ(x(k)) = 0.98sin(x1 (k)) . 0 25

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According to Lemma 1, we have   0 0 0   VHn = ±0.98 0 0 .   0 0 0

In simulation, the external disturbance signal is assumed to be d(k) = 0.8

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0.2sin(10k) and the fault is set up as  0.3sin(0.01k), 10 ≤ k ≤ 80 f (k) = 0, otherwise.

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Given γ = 1, σ = 0.3, ϑdh = π/1.5, ϑf l = π/10. By the proposed finite T frequency FD approach under ε1 = 0, ε2 = 0, ε3 = 1 and S = 0 0 0 , one can obtain the filter parameters  T L = −0.0439 0.0556 1.8674 , V = 3.4680

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and the optimal value on the fault sensitivity performance index is β =  T 1.3056. Let the initial conditions be x(0) = 0.02 −0.01 0.01 , xˆ(0) =  T 0 0 0 . The generated residual r(k) and its evaluation function are shown in Fig. 7. The release instants and release intervals are shown in Fig. 8. We observer that the fault can be easily detected and only 33 times are triggered. That is the proposed event-triggered FD approach provides 67% reduction in the data transmission. These results further demonstrate the effectiveness of the proposed method. It is worth noting that we fail to design a FD filter with the existing method. So the proposed method is less conservative than the existing one. 26

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Figure 8: Release interval with σ = 0.3.

6. Conclusions

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In this paper, we have investigated the problem of finite frequency H− /H∞ fault detection filter design under an event-triggered scheme for discrete-time Lipschitz nonlinear networked systems. Using a reformulated Lipschitz property combined with a novel lemma and slack variable techniques, a new filter design method has been proposed which makes full use of the frequency information of disturbances and faults to reduce design conservatism. The proposed method can be applied to systems even with large Lipschitz constants. The H− /H∞ filter design conditions are obtained in the formulation of LMIs. Finally, simulation results illustrate that the proposed design method can achieve a better performance than the existing full-frequency method in saving communication resources and detecting faults. Although only the inverted pendulum system has been studied in the simulation, the proposed FD scheme has great potentials to apply to a variety of engineering systems. It is encouraged to extend the presented results to more complex systems, such as stochastic dynamic systems [9, 45] and multi-agent systems [46, 47]. In addition, besides the proposed robust FD scheme, parameter-adaptation based FD scheme [48] and signal-based diagnosis scheme (e.g., spectrum analysis method and phase response curve method [49, 50]) will also be studied further in our future work.

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Acknowledgements

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This work was supported in part by the Funds of National Science of China (Grant Nos. 61420106016 and 61621004), and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant no. 2013ZCX01-01). References

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Ying Gu received the B.S. and M.S. degrees in mathematics from Northeast Normal University, China, in 2001 and 2005, respectively. Currently, she is pursuing the Ph.D. degree in control theory and control engineering from Northeastern University, China. Her research interests include fault detection, nonlinear systems and networked control systems.

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Guang-Hong Yang received the B.S. and M.S. degrees from Northeast University of Technology, Liaoning, China, in 1983 and 1986, respectively, and the Ph.D. degree in Control Engineering from Northeastern University, China (formerly, Northeast University of Technology), in 1994. He was a Lecturer/Associate Professor with Northeastern University from 1986 to 1995. He joined the Nanyang Technological University in 1996 as a Postdoctoral Fellow. From 2001 to 2005, he was a Research Scientist/Senior Research Scientist with the National University of Singapore. He is currently a Professor at the College of Information Science and Engineering, Northeastern University. His current research interests include fault-tolerant control, fault detection and isolation, non-fragile control systems design, and robust control. Dr. Yang is an Associate Editor for the International Journal of Control, Automation, and Systems (IJCAS), the International Journal of Systems Science (IJSS), the IET Control Theory & Applications, and the IEEE Transactions on Fuzzy Systems.

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