# Event-BasedH∞ Filter Design for Sensor Networks with Missing Measurements

Authors: Jinliang Liu; Shumin Fei; Engang Tian
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

In order to save network resources and network bandwidth, this paper proposed an event triggered mechanism based on sampled-data information, which has some advantages over existing ones. Considering the missing sensor measurements and the network-induced delay in the transmission, we construct a new event-basedH∞ filtering by taking the effect of sensor faults with different failure rates. By using the Lyapunov stability theory and the stochastic analysis theory, sufficient criteria are derived for the existence of a solution to the algorithm of the event-based filter design. Finally, an example is exploited to illustrate the effectiveness of the proposed method.


---

## Body

## 1. Introduction

The application of network technologies is becoming increasingly important in many areas for its predominant advantages (such as low cost, simple installation and maintenance, and high reliability). However, it is known that implementing a communication network can induce multiple channel transmission, packet dropout, and so on. This has motivated much attention to the research. Various techniques have been proposed to deal with the above issues, such as time triggered communication scheme [1, 2] and event triggered communication scheme [3–7]. In general, under a time triggered communication scheme, a fixed sampling interval should be selected under worse conditions such as external disturbances and time delay. However, such situation rarely occurs. Hence, time triggered communication scheme can lead to transmit much unnecessary information and inefficient utilization of limited network resources. Comparing with time triggered scheme, the event triggered scheme can save the network resources such as network bandwidth while maintaining the control performance. The adoption of the event triggered scheme has drawn a great deal of interest to the researchers. The authors in [3] firstly proposed a kind of event triggered scheme which decided whether the newly sampled signal should be transmitted to the controller and invested the controller design problem. In [8], the authors took the sensor and actuator faults into consideration and studied the reliable control design for networked control system under event triggered scheme. The authors in [9] were concerned with the control design problem of event triggered networked systems with both state and control input quantization. In [10], the authors discussed the event-based fault detection for the networked systems with communication delay and nonlinear perturbation.On the other hand, the filtering problem has been a hot topic over the past decades. A large number of outstanding results have been published [9, 11–18]. For example, the researchers in [9] studied the problem of event-based H∞ filtering for networked systems with communication delay. Most of them are based on an assumption that sensors are working without any flaws. However, the distortion of the sensor usually occurs due to the internal noise or external disturbance. Therefore, it is necessary to discuss the situation when the filter cannot receive the value of the process accurately. Fortunately, much effort has been put into this issue. The authors in [19] were concerned with reliable H∞ filter design for sampled-data systems with probabilistic sensor signal distortion. In [20], the authors investigated reliable H∞ filter design for T-S fuzzy model based networked control systems with random sensor failure.To the best of our knowledge, the filter design of event triggering network-based systems with random sensor failures is still an open problem, which motivates our present paper. The main contributions of the obtained results are as follows: (I) the insertion of the event triggering generator saves the network resources and network bandwidth. (II) A new kind of event triggering network-based systems with probabilistic sensor failures and network induced delay, which has not been investigated in the existing literatures, is proposed.This paper is outlined as follows. Section2 presents the modeling. Section 3 presents our main stability theorem and develops a filter design method. In Section 4, an example is given to illustrate the effectiveness of the proposed method.ℝ
n and ℝn×m denote the n-dimensional Euclidean space and the set of n×m real matrices; the superscript “T” stands for matrix transposition; I is the identity matrix of appropriate dimension; ∥·∥ stands for the Euclidean vector norm or the induced matrix 2-norm as appropriate; the notation X>0 (resp., X≥0), for X∈ℝn×n means that the matrix X is real symmetric positive definite (resp., positive semidefinite), when x is a stochastic variable. For a matrix B and two symmetric matrices A and C, [A*BC] denotes a symmetric matrix, where * denotes the entries implied by symmetry.


## 2. System Description

As shown in Figure1, our aim in this paper is to investigate an event-based reliable filtering design problem by taking the effect of sensor faults. Suppose the plant model is governed by
(1)x(k+1)=Ax(k)+Bw(k),y(k)=Cx(k),z(k)=Lx(k),
where x(k)∈ℝn is the state vector, y(k)∈ℝm is the measured output, z(k) is the signal to be estimated, w(k) is the process noise belonging to 𝕃2(0,∞), A, B, C, and L are known constant matrices with appropriate dimensions.Figure 1
The structure of an event triggered filter design system.Remark 1.
Considering the network induced delay, the transmission time of measured outputy(k) from sensor to filt cannot be neglected. The input of the filter is not y(k), but y^(k), in fact, y^(k)=y(k+τ(k)). τ(k) is the network induced delay, and τ(k)∈[0,τM), where τM is a positive real number.For the network-based system in described in Figure1, we propose the following filter:
(2)xf(k+1)=Afxf(k)+Bfy^(k),zf(k)=Cfxf(k),
where xf(k) is the filter state, y^(k) is the input of the filter, Af, Bf, Cf are the filter matrices of appropriate dimensions.If we take the missing sensor measurements into consideration, (2) can be described as
(3)xf(k+1)=Afxf(k)+BfΞy^(k),zf(k)=Cfxf(k),
where Ξ=diag⁡{Ξ1,Ξ2,…,Ξm}, Ξi∈[0,θ1](i=1,2,…) (θ1>1) being m unrelated random variables, and the mathematical expectation and variance of Ξi are αi and σi2, respectively.Remark 2.
Whenαi=1, it means the sensor i works normally. When αi=0, it means the sensor i completely failed and the signal transmitted by sensor i is lost. When Ξi∈[0,1], it means the signal at the filter is smaller or greater than it actually is [20].In order to reduce the load of network transmission and save the network resources such as network bandwidth, it is necessary to introduce an event triggered mechanism. As is shown in Figure1, an event generator is constructed between the sensor and filter, which is used to decide whether the measured output should be sent to the filter. We adopt the following judgement algorithm:
(4)[𝔼{Ξy(k)}-𝔼{Ξy(si)}]TΩ[𝔼{Ξy(k)}-𝔼{Ξy(si)}]≤σ[𝔼{Ξy(k)}]TΩ[𝔼{Ξy(k)}],
where Ω∈ℝm×m is a symmetric positive definite matrix, σ∈[0,1), Ξ=diag⁡{Ξ1,Ξ2,…}, and Ξi∈[0,θ1](i=1,2,…) are m unrelated random variables. Only when the expectation of a certain function of current sampled value y(k) and the previously transmitted one y(si) violate (4), it can be sent out to the filter.Remark 3.
Under the event triggering (4), the release times are assumed to be s0,s1,s2,…. Due to the delay in the network transmission, the measured output will arrive at the filter at the instants s0+τ(s0),s1+τ(s1),s2+τ(s2),…, respectively.Based on the above analysis, considering the behavior of ZOH, the input of the filter is(5)Ξy^(k)=Ξy^(si),k∈[si+τ(si),si+1+τ(si+1)-1].Similar to [4, 6, 11], for technical convenience, consider the following two cases.Case 1.
Whensi+1+τM≥si+1+τ(si+1)-1, define a function d(k) as
(6)d(k)=k-si,k∈[si+τ(si),si+1+τ(si+1)-1].
Obviously,
(7)τ(si)≤d(k)≤(si+1-si)+τ(si+1)-1≤1+τM.Case 2.
Whensi+1+τM≤si+1+τ(si+1)-1, consider the following two intervals:
(8)[si+τ(si),si+τM],[si+τM+l,si+τM+l+1].
Fromτ(k)≤τM, we can deduce that there must exist d satisfying
(9)si+d+τM<si+1+τ(si+1)-1≤si+d+1+τM.
Moreover,y(si) and y(si+l)l=1,2,…,d satisfy (4). Set
(10)I0=[si+τ(si),si+τM+1),Il=[si+τM+l,si+τM+l+1),Id=[si+d+τM,si+1+τ(si+1)-1],
where l=1,2,…,d-1. Clearly, we have
(11)[si+τ(si),si+1+τ(si+1)-1]=⋃i=0i=d‍Ii.
Defined(k) as
(12)d(k)={k-si,k∈I0,k-si-l,k∈Il,l=1,2,…,d-1,k-si-d,k∈Id.
Then, one can easily get
(13)τ(si)≤d(k)≤1+τM≜dM,k∈I0,τ(si)≤τM≤d(k)≤dM,k∈Il,l=1,2,…,d-1,τ(si)≤τM≤d(k)≤dM,k∈Id.
Due to si+1+τ(si+1)-1≤si+d+1+τM, the third row in (13) holds. Obviously,
(14)τ(si)≤τM≤d(k)≤dM,k∈Id.In Case1, for k∈[si+τ(si),si+1+τ(si+1)-1], define ei(k)=0. When it comes to Case 2, define
(15)Ξ¯ei(k)={0,k∈I0,Ξ¯y(si)-Ξ¯y(si+l),k∈Il,l=1,2,…,d-1,Ξ¯y(si)-Ξ¯y(si+d),k∈Id.It can be deduced from the definition ofΞ¯ei(k) and the event triggering scheme (4); for k∈[si+τ(si),si+1+τ(si+1)-1], the following inequality holds
(16)eiT(k)Ξ¯TΩΞ¯ei(k)≤σyT(k-d(k))Ξ¯TΩΞ¯y(k-d(k)).Remark 4.
From (15), it can be easily obtained that
(17)ei(k)={0,k∈I0,y(si)-y(si+l),k∈Il,l=1,2,…,d-1,y(si)-y(si+d),k∈Id.
Employingd(k)  ei(k), the input of the filter Ξy^(k) can be rewritten as
(18)Ξy^(k)=Ξy(si)=Ξ(y(k-d(k))+Ξei(k)),00000000k∈[si+τ(si),si+1+τ(si+1)-1].
Obviously,
(19)y^(k)=y(si)=(y(k-d(k))+ei(k)),000k∈[si+τ(si),si+1+τ(si+1)-1].Combining (19) and (3), we can get
(20)xf(k+1)=Afxf(k)+BfΞ(Cx(k-d(k))+ei(k)),zf(k)=Cfxf(k).Defineη(k)=[x(k)xf(k)], e(k)=z(k)-zf(k); the following filtering-error system based on (1) and (20) can be obtained as
(21)η(k+1)=A¯η(k)+Dx(k-d(k))+Dkx(k-d(k))+B¯ei(k)+B¯kei(k)+B¯1w(k),e(k)=L¯η(k),
where A¯=[A00Af], D=[0BfΞ¯C], Dk=[0Bf(Ξ-Ξ¯)C], B¯=[0BfΞ¯], B¯k=[0Bf(Ξ-Ξ¯)], B¯1=[B0], L¯=[L-Cf].Remark 5.
The event triggering scheme (4) can be applied to the situation when the sensor have failures. Besides, the effect of the network environment is also taken into consideration. From the modeling process, we can see that the system (21) is more general.Before giving the main results in the next section, the following lemmas will be introduced, which will be helpful in deriving the main results.Lemma 6 (see [21]).
For any vectorsx, y∈ℝn, and positive definite matrix Q∈ℝn×n, the following inequality holds:
(22)2xTy≤xTQx+yTQ-1y.Lemma 7 (see [22]).
Ω
1, Ω2, and Ω are matrices with appropriate dimensions, d(k)∈[0,dM]; then
(23)d(k)Ω1+(dM-d(k))Ω2+Ω<0,
if and only if the following two inequalities hold
(24)dMΩ1+Ω<0,dMΩ2+Ω<0.


## 3. Main Results

In this section, we will invest a new approach to guarantee the filter error system (21) to be globally asymptotically stable. A sufficient condition is established for (21). Then, the explicit filter design method in (20) is given.Theorem 8.
For given scalarsαi, μi(i=1,…,m), ρ∈[0,1), 0≤d(k)≤dM, and γ, under the event triggered communication scheme (4), the augmented system (21) is asymptotically stable with an H∞ performance index γ for the disturbance attention, if there exist positive definite matrices P, Q, R and matrices N, M with appropriate dimensions, such that
(25)Ω(s)=[Ω11+Γ+ΓT***Ω21Ω22**Ω310Ω33*Ω41(s)00-R]<0,s=1,2,
where
(26)Ω11=[PA¯+A¯P-2P+HTQH****DTP0***00-Q**B¯TP00-Ξ¯TΩΞ¯*B¯1TP000-γ2I],Γ=[NHM-N-M00],Ω21=[P(A¯-I)PD0PB¯PB¯1dMRH(A¯-I)dMRHD0dMRHB¯dMRHB¯1L¯00000σΩΞ¯C000],Ω22=diag⁡{-P,-R,-I,-Ω},Ω31=[0δ1PD^10000⋮0000δmPD^m000000δ1PB^10000⋮0000δmPB^m0],Ω41(1)=dMNT,Ω41(2)=dMMT,NT=[N1TN2TN3TN4TN5T],MT=[M1TM2TM3TM4TM5T],D^i=[0BfEiC],B^i=[0BfEi],(i=1,2,…,m),Ei=diag⁡{0,…,0︸i-1,1,0,…,0︸m-i},H=[I0].Proof.
Setδ(k)=x(k+1)-x(k), η¯(k)=η(k+1)-η(k); choose the Lyapunov functional candidate
(27)V(k)=ηT(k)Pη(k)+∑k-dMk-1xT(i)Qx(i)‍+∑i=-dM-1∑j=k+ik-1δT(j)Rδ(j).
Calculating the difference ofV(k) along the solution of (27) and taking the mathematical expectation, we obtain
(28)𝔼{ΔV(k)}=2ηT(k)P[(A¯-I)η(k)+Dx(k-d(k))00000000+B¯ei(k)+B¯1w(k)]+𝒜TP𝒜+∑i=1mσi2xT(k-d(k))[0BfEiC]T×P[0BfEiC]x(k-d(k))+∑i=1mσi2eiT(k)[0BfEi]TP[0BfEi]ei(k)+ηT(k)HTQHη(k)-xT(k-dM)Qx(k-dM)+𝔼{dMδT(k)Rδ(k)}-∑i=k-dMk-1δT(k)Rδ(k),
where 𝒜=(A¯-I)η(k)+Dx(k-d(k))+B¯ei(k)+B¯1w(k) and
(29)𝔼{dMδT(k)Rδ(k)}=𝔼{dMη¯T(k)HTRHη(k)¯}=dM𝒜THTRH𝒜.
Then by employing free weight matrix method [23, 24], we have
(30)2ξT(k)M[x(k-d(k))-x(k-dM)]-∑i=k-dMk-d(k)-1‍δ(i)=0,2ξT(k)N[x(k)-x(k-d(k))]-∑i=k-dMk-1‍δ(i)=0,
where ξT(k)=[ηT(k)xT(k-d(k))xT(k-dM)eiT(k)wT(k)]T.
By Lemma6, we can easily get
(31)-2ξT(k)M∑i=k-dMk-d(k)-1‍δ(i)≤(dM-d(k))ξT(k)MR-1MTξ(k)+∑i=k-dMk-d(k)-1‍δT(i)Rδ(i),(32)-2ξT(k)N∑i=k-dMk-1‍δ(i)≤d(k)ξT(k)NR-1NTξ(k)+∑i=k-d(k)k-1‍δT(i)Rδ(i).
Combine (28)–(31) and (16), we have
(33)𝔼{ΔV(k)}-γ2wT(k)w(k)+eT(k)e(k)≤ξT(k)[Ω11+Γ+ΓT]ξ(k)+(dM-d(k))ξT(k)MR-1MTξ(k)+d(k)ξT(k)NR-1NTξ(k)+𝒜TP𝒜+∑i=1mσi2xT(k-d(k))‍[0BfEiC]T×P[0BfEiC]x(k-d(k))+∑i=1mσi2eiT(k)‍[0BfEi]TP[0BfEi]ei(k)+σxT(k-d(k))CTΞ¯TΩΞ¯Cx(k-d(k))+dM𝒜THTRH𝒜+ηT(k)L¯TL¯η(k).
Subsequently, by the well known Schur complement and Lemma 7, from (25), we can deduce
(34)𝔼{ΔV(k)}-γ2wT(k)w(k)+eT(k)e(k)≤0.
Similar to the method in [25], the filter error system (21) is asymptotically stable.Based on Theorem8, a design method of the reliable filter in the form of (20) is given in Theorem 9.Theorem 9.
For given parametersα, σi(i=1,2,…,m), ρ∈[0,1), and 0≤d(k)≤dM, the filter error system (21) is asymptotically stable with H∞ performance level γ, if there exist positive definite matrices X, Q, R^, and Af, Bf, Cf, N10, N11, M10, M11, Mi, and Ni(i=2,3,4,5) with appropriate dimensions, such that
(35)Ω^(s)=[Ω^11+Γ^+Γ^T****Ω^21Ω^22***Ω^310Ω^33**Ω^4100Ω^44*Ω^51(s)Bw00-R]<0,s=1,2,(36)P1-P¯3>0,
where (37)Ω^11=[P1A+ATP1-2P1+Q*****P¯3A+A¯fT-2P¯3Af+AfT-2P3****CTΞ¯TB¯fTCTΞ¯TB¯fT0***000-Q**Ξ¯TB¯fTΞ¯TB¯fT00-Ξ¯TΩΞ¯*BTP1BTP¯3000-γ2I],Ω^21=[P1(A-I)A¯f-P¯3B¯fΞ¯C0B¯fΞ¯P1BP¯3(A-I)A¯f-P¯3B¯fΞ¯C0B¯fΞ¯P¯3B],Ω^22=[-P1*-P¯3-P¯3],Ω^31=[dMR(A-I)0000dMRBL-Cf00000σΩΞ¯C0000],Ω^41=[00δ1B¯fE1C00000δ1B¯fE1C00000⋮00000δmB¯fEmC00000δmB¯fEmC0000000δ1B¯fE100000δ1B¯fE100000⋮00000δmB¯fEm00000δmB¯fEm0],Γ^=[N100M10-N10-M1000N110M11-N11-M1100N20M2-N2-M200N30M3-N3-M300N40M4-N4-M400N50M5-N5-M500],Ω^33=diag⁡{-R,-I,-Ω},Ω^44=diag⁡{Ω^22,…,Ω^22︸2m},Ω51(1)=dMN¯T,Ω51(2)=dMM¯T,N¯T=[N10TN11TN2TN3TN4TN5T],M¯T=[M10TM11TM2TM3TM4TM5T].The filter parameters are given by
(38)Af=A¯fP¯3-1,Bf=B¯f,Cf=C¯fP¯3-1.Proof.
SinceP¯3>0, there exist nonsingular matrix P2 and symmetrical matrix P3>0 satisfying P¯3=P2TP3-1P2.
Define(39)P=[P1P2TP2P3],J=[I00P2TP3-1].
Now premultiply and postmultiply Equation (25) with Υ=diag⁡{J,I,I,…,I︸4,J,I,I,I,J,J,…,J︸2m,I} and ΥT, and define new variables as
(40)A¯f=A^fP¯3,A^f=P2TAfP2-T,B¯f=P2TBf,C¯f=C^fP¯3,C^f=CfP2-T,N1TJT=[N¯10TN¯11T],M1TJT=[M¯10TM¯11T].
We can obtain (35). Therefore, (35) holds, only if (25) holds. From Theorem 8, the filter error system (21) is asymptotic stable with H∞ performance level γ.Similar to the analysis of [25], the filter parameters in (20) can be obtained as (38).


## 4. Simulation Examples

Consider a specific network controlled system of Equation (21) under a structure:
(41)A=[0.10.4-0.40.1],B=[-0.70.2],C=[01],L=[11].
Assume 0≤d(k)≤4 and the failure rates of the sensors are α1=0.8 and σ1=0.05.According to Theorem9, when H∞ performance level γ=0.8, the following parameters can be obtained from the solution of (35) and (36) by using the LMI technique:
(42)P¯3=[0.75210.58010.58011.4216],A¯f=[-0.08370.3567-0.32380.2977],B¯f=[-0.0036-0.0019],C¯f=[-0.6268-0.6181].
From (38), the corresponding filter parameters can be obtained as
(43)Af=[-0.44480.4325-0.86410.5620],Bf=[-0.0036-0.0019],Cf=[-0.7268-0.1382]
and the parameter in the event triggering scheme (4) is Ω=0.0239.Suppose the initial conditionx(0)=[0.20.1]T and external disturbance
(44)w(k)={0.055s≤k≤15s0else.Based on the designed filter above, the response of the errore(k) and the probabilistic failure Ξ are given in Figures 2 and 3, respectively. Figure 4 describes the release instants and release interval. It is easy to see from Figures 2–4 that the filter design method in this paper is effectiveness.Figure 2
The response of the errore(k).Figure 3
The probabilistic failureΞ.Figure 4
The release instants and release interval.


## 5. Conclusion

This paper investigates aH∞ filter design for a class of network-based systems under an event triggered mechanism. In particular, the system under study is a more general sensor failure model. Considering the uncertain time delay, the uncertain network environment and probabilistic missing sensor measurements, we introduce an event triggered mechanism into the system. By using the free-weighting matrix method and the LMI techniques, the fundamental stability conditions are obtained and the filter design methods are developed. Finally, a numerical example is given to demonstrate the effectiveness of the proposed designed method.We would like to point out that it is possible to extend our main results to the nonlinear systems such as T-S fuzzy systems, and complex network systems. This will also be one of our future research issues.



---

*Source: 512037-2014-02-24.xml*