# DistributedH∞ Consensus Filtering for Parabolic Distributed Parameter Systems with Multiple Missing Measurements: A Mobile Sensing Approach

Authors: Xueming Qian; Baotong Cui
Journal: Journal of Control Science and Engineering
 (2023)

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## Abstract

This paper investigates the distributedH∞ consensus filtering issue for a class of distributed parameter systems with bounded disturbance. In a framework of optimizing performance, a new approach to improving filter performance is proposed by employing mobile sensor networks. Moreover, the information missing in mobile sensor networks is modeled as a conditional probability distribution. The aim of the filtering challenge is to construct a distributed consensus filter such that the filtering error system is globally asymptotically stable in the mean square, and what disturbances do to the estimation accuracy is attenuated at the H∞ consensus performance level. Utilizing the Lyapunov direct approach and the spatial operator technique, several sufficient criteria are given for the proposed filter to satisfy the H∞ consensus performance constraint. Finally, a numerical simulation is given to demonstrate the effectiveness of the design scheme of the proposed filter.


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## Body

## 1. Introduction

Over the last decade or so, wireless sensor networks (WSNs) have been successfully applied in a wide range of areas, such as environmental monitoring, military applications, smart buildings, and health management [1]. A sensor network, in general, comprises a group of sensing devices that can communicate wirelessly to coordinate their task and collect data. Sensor network observations are required to be processed in order to ensure decision-making. As a result, the problem of state estimation for a wide variety of systems has attracted considerable attention. A centralized data processing method can be used to estimate the state of a system, determine its parameters, and identify its detection sources. To study the state estimation problem of discrete linear systems with Markovian delay and packet losses, a class of limited received history estimators based on the idea of jump linear estimators was proposed in [2].Yet each sensing node is equipped with an embedded device which has a limited power. The use of distributed estimation is an effective way to save energy. A sensor scheduling strategy was proposed in [3] based on energy conservation while focusing on distributed state estimation for wireless sensor networks under energy constraints. In [4], a robust performance-preserving state estimator was designed to obtain state estimation of discrete-time complex networks with estimation errors satisfying mean-square exponential convergence for real networks with packet losses and noise interference in data transmission. To further extend the estimation accuracy in case of data missing, H∞ estimation is considered. An H∞ estimator was designed in [5], which achieved the state estimation of a class of time-varying neural networks under measurement degradation and randomly occurring deception attacks, such that the predefined probabilistic constraints of error dynamics were satisfied and that H∞ performance was guaranteed. It is noted that the systems to which these state estimates are applied are lumped parameter systems.Also, the estimation of distributed parameter systems is receiving more attention recently. The work to predict the water quality concentration distribution in the water supply appeared in [6], where the system was modeled by the reaction diffusion equation. For distributed parameter processes [7], optimal state estimators are provided based on optimal sensor locations. An adaptive consensus filter was proposed in [8] for filtering in distributed parameter systems with fixed sensors located in the system in a spatially distributed way.Data collected by a network of mobile sensors are another type of estimate in spatially distributed systems. In [9], an output feedback controller was designed using a mobile sensor network with actuators to stabilize a class of randomly distributed parameter systems faster. With the introduction of this mobile actuator-sensor network, the control performance of the spatially distributed process is enhanced, and thus, an optimization framework for the control problem of distributed parameter systems is built. In [10], the estimation problem of spatially distributed processes described by a partial differential equation was studied using a group of sensors which could move in the spatial domain in order to enhance the performance of the state estimator. Compared to a group of immobile sensors, mobile sensors have the flexibility to move around when collecting data. Therefore, the number of sensors in the system can be reduced, which reduces energy consumption and increases efficiency. The state estimator designed with the guidance of mobile sensors is capable of achieving better estimating performance as shown in [10]. Related research work has been further extended to other aspects. In [11], an accuracy reconstruction of the traffic flow was achieved using data collected by a mobile sensor network for the Lighthill–Whitham–Richards model described by partial differential equations. And the parameter estimation of distributed processes by optimizing sensor locations was studied in [12].Unfortunately, it is often the case that information regarding the system states is only partially available in practice. In real-world engineering, sensor temporal failure or network transmission delay are the main causes of the missing measurement phenomenon in networked environments. The first study addressing the filtering problem with missing measurements defined by Bernoulli distributions appeared in [13], and works addressing other lumped parameter systems with probabilistic missing data followed in [14–16]. The studies mentioned above brought up the fact that the observed output from fixed sensors was incomplete. In the measured output produced by mobile sensor networks, missing data have, however, often been ignored up to this point. A potential challenge of this issue is that the coefficients of the evolution equation which represent the spatial distribution process are operators and need to be dealt with by functional analysis.As was previously stated, few studies have been undertaken to deal with distributedH∞ consensus filtering systems with missing measurements. Our motivation for writing this paper is to focus on the problem of distributed H∞ consensus filtering for parabolic distributed parameter systems with multiple missing measurements utilizing mobile sensor networks. Our approach aims to derive the state of spatially dispersed processes from the output with missing data in such a way that the filtering error reaches zero asymptotically stable in the mean square and that a certain H∞ perturbation rejection attenuation level is ensured. For each moving sensing device, optimal trajectory planning can be produced with the use of spatial operators and the Lyapunov direct method. After which, a distributed filter subject to H∞ consensus performance is constructed while taking into account the use of mobile sensors to improve the filter’s performance. To demonstrate the effectiveness on the proposed conditions, the simulation is offered as an example.


## 2. Problem Formulation

In chemical reactors, parabolic partial differential equations can often be used to model changes in concentration in chemical reactions and the evolution of temperature in heat exchange [17]. The spatial distribution of temperature in a chemical reactor can be described by the following parabolic partial differential equation:(1)∂Lt,η∂t=∂∂ηaη∂Lt,η∂η−φLt,ηLt,η+dηwt,zt=∫0lbηLt,ηdη,subject to the initial condition(2)L0,η=L0η,and having the Dirichlet boundary condition as(3)Lt,0=Lt,l=0,t≥0,where Lt,η indicates that the system is in the state of the location η at the time t and η is a spatial variable, varying in Ω=0,l. The other variable of the state is the time t∈0,+∞.The diffusion coefficient aη≥a0>0. φLt,η is a nonlinear function and satisfies φm≤φLt,η≤φM, where φm and φM are known bounds. wt is an external disturbance, and dη denotes its spatial distribution. zt is the output to be estimated, and bη denotes the output’s spatial distribution.The spatial measurements fromn mobile sensors are as follows:(4)yt=y1ty2t⋮ynt=β1t∫0lc1η;η1stLt,ηdηβ2t∫0lc2η;η2stLt,ηdη⋮βnt∫0lcnη;ηnstLt,ηdη,or(5)yit=βit∫0lciη;ηistLt,ηdη,i=1,2,…,n,where yit is the measurement result that the ith mobile sensor collected. The spatial distribution of the ith moving sensing device is given by a nonnegative bounded function ciη;ηist, where ηist is the position function of the ith sensor with respect to the time t, varying in 0,l. It should be noticed that ciη;ηist indicates that each sensor may have a different distribution.The stochastic variableβit∈R,i=1,2,…,n has values 1 and 0 for every i with(6)Probβit=1=β¯i,and Probβit=0=1−β¯i,where β¯i,i=1,2,…,n are given constants taking values in [0, 1].It is assumed thatβit is independent of the system’s initial state and is also independent of wt. Hence, it is easy to draw the following results:(7)Eβit−β¯i=0as well asEβit−β¯i2=β¯i1−β¯i.Remark 1.
The distributed parameter control system in this paper receives the measurement output from the mobile sensori. Probabilistic missing data are unavoidable in sensor networks as sensing nodes can move around, and there is only so much channel bandwidth for signal transmission. The new sensor network model includes missing data at random and can more accurately depict mobile sensor network reality. Note that such a data missing mode was introduced in [13] first by Nahi.
As a general matter, the spatial distribution of mobile sensors at each time-varying positionηist can be expressed as(8)ciη;ηis=ciη,ifη∈ηis−εi,ηis+εi,0,otherwise,or(9)ciη;ηist=ciηHη−ηis−εi−Hη−ηis+εi,which are depicted as two different heaviside step functions.Remark 2.
Nowadays, most of the spatial distributions [8] about mobile agents are described by the following expression:(10)cη;ηis=1,ifη∈ηis−ε,ηis+ε,0,otherwise.
The aforementioned expression of the assumption indicates that the network of sensors is homogeneous. And the distribution of sensors given in (8) shows that each moving sensing device in the mobile sensor network may have a different location distribution; that is, ciη;ηist denotes a nonhomogeneous network. Thus, since Assumption (8) is employed, the result of this study would be less conservative.
More generally, the distribution of each sensor also can be piecewise smooth in local. For instance, one sensor’s distribution in a symmetric intervalηis−ε,ηis+ε can be shown as(11)ciη=1ε,ifη∈ηis−ε3,ηis+ε3,12ε,ifη∈ηis−ε2,ηis−ε3∪ηis+ε3,ηis+ε2,0,ifη∈ηis−ε,ηis−ε2∪ηis+ε2,ηis+ε.
Therefore, the more general distribution of each mobile sensor can be taken into account within the proposed framework as follows:(12)ciη;ηis=∑j=1ncijηHη−ηi0s+j−1Δh−Hη−ηi0s+jΔh,where Δh=ε++ε−/m2, ηi0s=ηis−ε−,i,j=1,2,…,n.
It is advantageous to rewrite the parabolic system (1) in an abstract way in order to apply the Lyapunov direct method for H∞ performance analysis of the filtering error system and the optimal moving strategy of mobile sensors.
Letℋ be a Hilbert space that has the inner product ⋅,⋅, and its induced norm is ⋅. Consider that ℬ is a reflective Banach space, which is continuous and densely embedded in ℋ, ⋅ as a norm of ℬ. ℬ∗ is ℬ’s conjugated dual, whereas ⋅∗ is its induced norm. It follows ℬ⟶ℋ⟶ℬ∗ with both continuously and embedding dense, and as a result, we have h≤αh,h∈ℬ, where α [18] is positive constant.
Here, the linear operatorA:ℬ⟶ℬ∗ which is called state operator satisfying the following assumptions [18]:(A1)h,Ag≤ϱ0hg, h,g∈ℬ, where ϱ0>0, namely, A is bounded.(A2)h,−Ah≥a0h2, h∈ℬ, where a0>0, namely, −A is coercive.Aside from that, the perturbation operatorD:R⟶ℬ∗ can be defined as(13)Dw,h=∫0ldηwthηdη,or,Dwt=dηwt.Satisfying the following assumption:(A3)D is bounded, that is, h,Dh≤dh,h, where h∈ℬ.In a similar way, the operatorℬ:R⟶ℬ∗ is offered by(14)ℬh,g=∫0lbηhηgηdη.with the underlying assumption:(A4)h,ℬh≤σbh,h, that is, ℬ is bounded.
Then, the distributed parameter system (1) can be rewritten to compact form as follows:(15)L˙t=ALt+Dwt,zt=ℬLt,where ℋ=L2Ω is the state space. The current state of the system is Lt,⋅=Lt,η:0≤η≤l. The Sobolev space ℬ=H010,l=g∈H1Ω|g0=gl=0 rules over the space ℬ, and its conjugate dual space is ℬ∗=H−1Ω.
Let the infinitesimal operatorA=d/dηaηd/dη−φ, and its proposed domain is established by DA=g∈L2Ω:g,g′are absolutely continuous,g″∈L2Ωandg0=gl=0. The domain DA of the operator A is dense in ℋ and thus generates a strongly continuous semigroup Tt,t≥0 [18].Remark 3.
The following calculation can be employed to verify that the bounded and coercivity assumptions (A1) and (A2) are valid:(16)h,Ag=∫0lddηaηdhηdη−φhηhηgηdη≤a0∫0ldhηdηdgηdηdη+φM∫0lhηgηdη≤a0∫0ldhηdη2dη∫0ldgηdη2dη+φM∫0lh2ηdη∫0lg2ηdη=a0hg+φMhg≤a0hg+φMα2hg=ϱ0hg,where ϱ0=a0+φMα2. The bounded assumption isestablished. For coercivity, you can see the following proof:(17)h,−Ah=∫0l−ddηaηdhηdη−φhηhηhηdη≥a0∫0ldhηdη2dη+φm∫0lh2ηdη=a0h2+φmh2≥a0h2.
In the same way, the measurement output (4) is also expressed as(18)yt=ΛβtCηstLt,where Λβt=diagβ1t,β2t,…,βnt and yt=y1t,y2t,…,yntT. The operator Cηst:ℬ⟶R×R×⋯×R⏟n which described the output is given by(19)Cηsth,g=∫0lc1η;η1sthηgηdη∫0lc2η;η2sthηgηdη⋮∫0lcnη;ηnsthηgηdη,where the vector of the sensor time-varying location function as a parameter is then expressed as ηst=η1st,η2st,…,ηnstT. In fact, as nonnegative distribution ciη;ηisti=1,2,…,n is bounded, Cηst satisfies the following assumption:(A5)Cηst is bounded, that is, h,Cηsth≤σch,h.
One can directly see that the observation operator is self-adjoint and that its norm is determined by the embedding constantα and the measure of the spatial domain μΩ.
In this paper, the state of the target (1) is estimated by taking into account a mobile sensor network with multiple missing data. Here, the following filter structure for mobile sensors i is given:(20)L^˙it=AL^it+C∗ηistγiyit−β¯iCηistL^it−Gi∑j≠iL^it−L^jt,z^it=ℬL^it,where L^it denotes the state estimation of ith mobile sensor, z^it denotes the estimation of zt after applying the filter, yit is the output measurement of the ith moving sensing device, that is, yit=βitCηistLt, and C∗ηist is the adjoint of the observation operator Cηist. Observer gains are denoted by γi>0, whereas consensus filter gains are denoted by Gi. Furthermore, L^i0=L^i0≠L0for all i=1,2,…,n.
Lettingeit=Lt−L^it and z˜it=zt−z^it, (15) and (20) can be employed to derive the filtering error system as follows:(21)e˙it=Acηisteit−C∗ηistγiβit−β¯iCηistηt+Gi∑j≠iejt−eit+Dwt,z˜it=ℬeit,where Acηist=A−C∗ηistγiβ¯iCηist, also with ei0=L0−L^i0≠0.
Given the fact thatCηist is self-adjoint, it implies that the closed-loop operator Acηist is self-adjoint. Also, Acηist is invertible when (A1), (A2), and (A5) are combined.
Our goal in solving the issue raised is to build a distributedH∞ consensus filter that takes the form of (20) so as to achieve the following conditions:(1)
The filtering error system withwt=0 has a zero solution that is globally asymptotically stable in the mean square.(2)
For the provided disturbance attenuation levelγ>0, under the zero initial condition, the H∞ consensus performance constraint fits the following inequality:(22)1n∑i=1nz˜it2≤γ2wt2.For the purpose of obtaining meaningful results, the definition and lemmas stated below have been introduced.Definition 1.
The filtering error system (21) with wt=0 is said to be globally asymptotically stable in the mean square if(23)limt⟶+∞Eeit2=0,which holds for any i∈1,2,…,n.Definition 2.
The filters (20) are said to be distributed H∞ consensus filters if their filtering error z˜it satisfy the following inequalities:(24)1n∑i=1nz˜it2≤γ2wt2,where the disturbance attenuation level γ>0 is given, for any i∈1,2,…,n.Remark 4.
The average filtering error ought to satisfy theH∞ performance constraint in the sensor network when the value of the filtering error z˜it,i=1,2,…,n satisfies the H∞ consensus performance constraint. Moreover, the H∞ consensus performance constraint will reduce the H∞ performance constraint if only one sensor exists in the network.Lemma 1.
Assuming thatϵ be a positive scalar and that v1,v2 be any n-dimensional real vector. The following inequality is thus true:(25)2v1,v2≤ϵv1,v1+ϵ−1v2,v2.Lemma 2.
(Barbalat’slemma [19]). In the scenario that a nonnegative function ft is the Lebesgue integer and uniformly continuous on 0,+∞, then limt⟶+∞ft=0.The corresponding theorem contains the primary results of this work.


## 3. Main Results and Proofs

### 3.1. Stability Analysis

Theorem 1.
The spatial distribution of the mobile sensors (12) and the consensus filter (20) are given. If there exist two positive constants pi and qi such that the following inequalities hold:(26)qi≥12,pi≥γi2β¯i1−β¯iσc4α24a0,and the following is the mobile sensor velocity law: (27)η˙ist=−ρiγiβ¯i∑j=1nWij1+κi∑k≠i∑j=1nWkj2,where(28)Wij=∫ηi0s+j−1Δhηi0s+jΔh∂cijη∂ηcijηei2t,ηdη+cij2ηi0s+j−1Δh+0ei2t,ηi0s+j−1Δh−cij2ηi0s+jΔh−0ei2t,ηi0s+jΔh,i,j,k=1,2,…,n,with Δh=ε++ε−/m, ηi0s=ηis−ε−, and ρi>0 denotes the velocity gain of ith mobile sensor; the zero solution of a filtering error system (21) with wt=0 is globally asymptotically stable in the mean square. The mobile sensing scheme improves the filter performance by accelerating the convergence of the filtering error eit to zero.Proof.
It is simple to verify the closed-loop operatorAcηist from (A1) and (A2), which satisfies the following criteria.(29)∑i=1nh,Acηistg=∑i=1nh,A−C∗ηistγiβ¯iCηistg≤ϱ0nhg+h,C∗ηstΓΛ¯βCηstg=ϱ0nhg+Λ¯βΓCηsth,Cηstg≤ϱ0nhg+λmaxΛ¯βΓα2μ2Ωhg=ϱhg,where ϱ=ϱ0n+λmaxΛ¯βΓα2μ2Ω>0, Λ¯β=diagβ¯1,β¯2,…,β¯n, Γ=diagγ1,γ2,…,γn, and Cηst=αμΩ.(30)∑i=1nh,−Acηisth=∑i=1nh,−A−C∗ηistγiβ¯iCηisth≥a0nh2+h,C∗ηstΓΛ¯βCηsth=a0nh2+Λ¯βΓCηsth,Cηsth≥a0nh2+λminΛ¯βΓCηsth2>a0nh2.
Considering the following parameter-dependent Lyapunov functional, we obtain(31)Vt=−∑i=1neit,Acηisteit+∑i=1nLt,piLt.
As defined byℒVt=limΔ⟶0+EVt+Δ|t−Vt/Δ, by applying the dynamics of the filteringerror (21), we obtain ℒV that(32)ℒVt=−∑i=1nEe˙it,Acηisteit−∑i=1nEeit,Acηiste˙it−∑i=1nEeit,dAcηistdteit+∑i=1nL˙t,piηt+∑i=1nηt,piL˙t.
It was simple to arrive at the following conclusion by taking into account (6) and (7) and noting that Acηist is self-adjoint.(33)−∑i=1nEe˙it,Acηisteit−∑i=1nEeit,Acηiste˙it=−2∑i=1nEAcηisteit−C∗ηistγiβit−β¯iCηistLt+Gi∑k≠iekt−eit,Acηisteit=−2∑i=1nEAcηisteit,Acηisteit+2∑i=1nEC∗ηistγiβit−β¯iCηistLt,Acηisteit−2∑i=1nEGi∑k≠iekt−eit,Acηisteit.
Observing the assumption (A5) and Lemma1, the following is true:(34)2∑i=1nEC∗ηistγiβit−β¯iCηistLt,Acηisteit≤∑i=1nqiEC∗ηistγiβit−β¯iCηistLt,C∗ηistγiβit−β¯iCηistLt+∑i=1nqi−1EAcηisteit,Acηisteit=∑i=1nqiγi2β¯i1−β¯iC∗ηstCηstLt2+∑i=1nqi−1EAcηisteit,Acηisteit≤∑i=1nqiγi2β¯i1−β¯iσc4Lt2+∑i=1nqi−1EAcηisteit,Acηisteit,for any scalar qi>0i=1,2,…,n.
Choosing the consensus filter here results inGi=Ac−1ηist for simplicity, and we have(35)−2∑i=1nEGi∑k≠iekt−eit,Acηisteit=−2∑i=1n∑k≠iekt−eit,eit=−2∑k≠iekt−eit,ekt−eit=−2∑k≠iekt−eit2.
Substituting (34)-(35) into (36) leads to(36)−∑i=1nEe˙it,Acηisteit−∑i=1nEeit,Acηiste˙it=∑i=1n−2+qi−1EAcηisteit2+∑i=1nqiγi2β¯i1−β¯iσc4Lt2−2∑k≠iekt−eit2.
Then, the third term of (32) has(37)−∑i=1nEeit,dAcηistdteit=∑i=1nEeit,ddtC∗ηistγiβ¯iCηisteit=2∑i=1nγiβ¯iCηisteit,η˙istdCηistdηeit=2∑i=1nη˙istγiβ¯i∫0lciη;ηist∂ciη;ηist∂ηei2t,ηdη=2∑i=1nη˙istγiβ¯i∑j=1n∫ηi0s+j−1Δhηi0s+jΔhcijη∂∂ηcijηHη−ηi0s+j−1Δh−Hη−ηi0s+jΔhei2t,ηdη=2∑i=1nη˙istγiβ¯i∑j=1n∫ηi0s+j−1Δhηi0s+jΔh∂cijη∂ηHη−ηi0s+j−1Δh−Hη−ηi0s+jΔh+cijηδη−ηi0s+j−1Δh−δη−ηi0s+jΔhcijηei2t,ηdη=2∑i=1nη˙istγiβ¯i∑j=1nWij.Here, Wij,i,j=1,2,…,n are defined in Theorem 1.
The selection(38)η˙ist=−ρiγiβ¯i∑j=1nWij,yields in (37) negative definite, where ρi>0i=1,2,…,n is the diagonal entries of the positive definite diagonal matrix Λρ. It is worth noting that η˙ist indicates the velocity of each mobile sensing device.
Moreover, the choice of the velocity law for the mobile sensing device is decoupled, meaning that it is only taken into account for its own measurement. And in a real mobile sensor network, not only its velocity depends on itself, but also its neighbors contribute to its velocity law. Every mobile sensor (38), regardless of its velocity law, can be modified to(39)η˙ist=−ρiγiβ¯i∑j=1nWij1+κi∑k≠i∑j=1nWkj2,where κi>0,i,j,k=1,2,…,n.
Calculating the derivative of the second part of the Lyapunov function yields(40)∑i=1nL˙t,piηt+∑i=1nηt,piL˙t=2∑i=1npiLt,ALt≤−2∑i=1npia0Lt2≤−2∑i=1npia0α2Lt2.
By substituting (36), (39)–(40) into (32), we obtain(41)ℒVt≤∑i=1n−2+qi−1EAcηisteit2+∑i=1nqiγi2β¯i1−β¯iσc4−2pia0α2Lt2−2∑k≠iEekt−eit2−2∑i=1nρiγi2β¯i2∑j=1nWij21+κi∑k≠i∑j=1nWkj2.
The inequalities (26) are feasible, which implies ℒVt≤0.
Thus, it is easy to deduce from (41) and the embedding that(42)ℒVt≤−∑i=1nEAcηisteit2≤−α0∑i=1nEeit2,where α0>0 is a constant that contains the embedding constant α and the coercivity constant a0. By employing the Ito^ formula, combined with (42), we can obtain(43)EVt=EV0+∫0tℒVsds≤EV0−α0∫0t∑i=1nEeit2ds,which implies that(44)∫0t∑i=1nEeit2ds≤1α0V0.
In addition, it can confirm that∑i=1nEeit2 is uniformly continuous on 0,+∞. Consequently, it follows from Lemma 1 that(45)limt⟶+∞Eet2=0.
The proof is complete if the filtering error system (21) with wt=0 is globally asymptotically stable in the mean square, as stated in Definition 1.Remark 5.
In theory, if we select the Lyapunov functional as(46)Vt=−ert∑i=1neit,Acηisteit+ert∑i=1nLt,piLt,then the filtering error system (21) with wt=0 can also be proved to be globally exponentially stable in the mean square as the similar way in Theorem 1.


### 3.2.H∞ Consensus Performance Analysis

Next, we focus on analyzing theH∞ performance of the filtering error system (21) under the zero initial condition.Theorem 2.
The disturbance attenuation levelγ>0 and the filter parameter γi and Gi are given. Under the assumption (A1)–(A5), the zero solution of the filtering error system (21) with wt=0 is globally asymptotically stable in the mean square. The H∞ consensus performance (22) is achieved for all nonzero wt if, with the initial condition, such that the following matrix inequality holds:(47)Ψ=−α0+σb2dϱdϱ−γ2<0.Proof.
(42) implies Ψ<0, which is simple to prove. In consideration of Theorem 1, the filtering error system (21) is globally asymptotically stable in the mean square. Let us now concentrate on how the closed-loop system performs in terms of the H∞ consensus. Likewise, as in Theorem 1, we build the Lyapunov functional candidate Vt. The same line computation used in Theorem 1 results in(48)ℒVt≤−α0∑i=1nEeit2+2∑i=1nEDwt,−Acηisteit≤∑i=1nEζt,Ψ^ζt,where ζt=et,wtT and Ψ^=−α0dϱdϱ0.
In order to address theH∞ consensus performance of the system (21), we provide(49)J=E∑i=1n∫0Tz˜it2−γ2wt2dt=E∑i=1n∫0Tz˜it,z˜itdt−nγ2∫0Twt,wtdt.
For any nonzero external disturbanceswt from (47) to (48), under the zero initial condition, we obtain(50)J≤E∑i=1n∫0Tfz˜it,z˜it−γ2wt,wt+ℒVtdt≤E∑i=1n∫0Tfℬeit,ℬeit−γ2wt,wt+∑i=1nEζt,Ψ^ζtdt≤∑i=1n∫0TfEζt,Ψζtdt,where Ψ is defined in Theorem 2.
Along the same lines as in the argument of Theorem1, we can establish that J<0. Letting Tf⟶∞, we have(51)∑i=1n∫0Tz˜it,z˜itdt<nγ2∫0Twt,wtdt,namely,(52)1n∑i=1nz˜it2<γ2wt2.
The theorem’s proof here is completed.


## 3.1. Stability Analysis

Theorem 1.
The spatial distribution of the mobile sensors (12) and the consensus filter (20) are given. If there exist two positive constants pi and qi such that the following inequalities hold:(26)qi≥12,pi≥γi2β¯i1−β¯iσc4α24a0,and the following is the mobile sensor velocity law: (27)η˙ist=−ρiγiβ¯i∑j=1nWij1+κi∑k≠i∑j=1nWkj2,where(28)Wij=∫ηi0s+j−1Δhηi0s+jΔh∂cijη∂ηcijηei2t,ηdη+cij2ηi0s+j−1Δh+0ei2t,ηi0s+j−1Δh−cij2ηi0s+jΔh−0ei2t,ηi0s+jΔh,i,j,k=1,2,…,n,with Δh=ε++ε−/m, ηi0s=ηis−ε−, and ρi>0 denotes the velocity gain of ith mobile sensor; the zero solution of a filtering error system (21) with wt=0 is globally asymptotically stable in the mean square. The mobile sensing scheme improves the filter performance by accelerating the convergence of the filtering error eit to zero.Proof.
It is simple to verify the closed-loop operatorAcηist from (A1) and (A2), which satisfies the following criteria.(29)∑i=1nh,Acηistg=∑i=1nh,A−C∗ηistγiβ¯iCηistg≤ϱ0nhg+h,C∗ηstΓΛ¯βCηstg=ϱ0nhg+Λ¯βΓCηsth,Cηstg≤ϱ0nhg+λmaxΛ¯βΓα2μ2Ωhg=ϱhg,where ϱ=ϱ0n+λmaxΛ¯βΓα2μ2Ω>0, Λ¯β=diagβ¯1,β¯2,…,β¯n, Γ=diagγ1,γ2,…,γn, and Cηst=αμΩ.(30)∑i=1nh,−Acηisth=∑i=1nh,−A−C∗ηistγiβ¯iCηisth≥a0nh2+h,C∗ηstΓΛ¯βCηsth=a0nh2+Λ¯βΓCηsth,Cηsth≥a0nh2+λminΛ¯βΓCηsth2>a0nh2.
Considering the following parameter-dependent Lyapunov functional, we obtain(31)Vt=−∑i=1neit,Acηisteit+∑i=1nLt,piLt.
As defined byℒVt=limΔ⟶0+EVt+Δ|t−Vt/Δ, by applying the dynamics of the filteringerror (21), we obtain ℒV that(32)ℒVt=−∑i=1nEe˙it,Acηisteit−∑i=1nEeit,Acηiste˙it−∑i=1nEeit,dAcηistdteit+∑i=1nL˙t,piηt+∑i=1nηt,piL˙t.
It was simple to arrive at the following conclusion by taking into account (6) and (7) and noting that Acηist is self-adjoint.(33)−∑i=1nEe˙it,Acηisteit−∑i=1nEeit,Acηiste˙it=−2∑i=1nEAcηisteit−C∗ηistγiβit−β¯iCηistLt+Gi∑k≠iekt−eit,Acηisteit=−2∑i=1nEAcηisteit,Acηisteit+2∑i=1nEC∗ηistγiβit−β¯iCηistLt,Acηisteit−2∑i=1nEGi∑k≠iekt−eit,Acηisteit.
Observing the assumption (A5) and Lemma1, the following is true:(34)2∑i=1nEC∗ηistγiβit−β¯iCηistLt,Acηisteit≤∑i=1nqiEC∗ηistγiβit−β¯iCηistLt,C∗ηistγiβit−β¯iCηistLt+∑i=1nqi−1EAcηisteit,Acηisteit=∑i=1nqiγi2β¯i1−β¯iC∗ηstCηstLt2+∑i=1nqi−1EAcηisteit,Acηisteit≤∑i=1nqiγi2β¯i1−β¯iσc4Lt2+∑i=1nqi−1EAcηisteit,Acηisteit,for any scalar qi>0i=1,2,…,n.
Choosing the consensus filter here results inGi=Ac−1ηist for simplicity, and we have(35)−2∑i=1nEGi∑k≠iekt−eit,Acηisteit=−2∑i=1n∑k≠iekt−eit,eit=−2∑k≠iekt−eit,ekt−eit=−2∑k≠iekt−eit2.
Substituting (34)-(35) into (36) leads to(36)−∑i=1nEe˙it,Acηisteit−∑i=1nEeit,Acηiste˙it=∑i=1n−2+qi−1EAcηisteit2+∑i=1nqiγi2β¯i1−β¯iσc4Lt2−2∑k≠iekt−eit2.
Then, the third term of (32) has(37)−∑i=1nEeit,dAcηistdteit=∑i=1nEeit,ddtC∗ηistγiβ¯iCηisteit=2∑i=1nγiβ¯iCηisteit,η˙istdCηistdηeit=2∑i=1nη˙istγiβ¯i∫0lciη;ηist∂ciη;ηist∂ηei2t,ηdη=2∑i=1nη˙istγiβ¯i∑j=1n∫ηi0s+j−1Δhηi0s+jΔhcijη∂∂ηcijηHη−ηi0s+j−1Δh−Hη−ηi0s+jΔhei2t,ηdη=2∑i=1nη˙istγiβ¯i∑j=1n∫ηi0s+j−1Δhηi0s+jΔh∂cijη∂ηHη−ηi0s+j−1Δh−Hη−ηi0s+jΔh+cijηδη−ηi0s+j−1Δh−δη−ηi0s+jΔhcijηei2t,ηdη=2∑i=1nη˙istγiβ¯i∑j=1nWij.Here, Wij,i,j=1,2,…,n are defined in Theorem 1.
The selection(38)η˙ist=−ρiγiβ¯i∑j=1nWij,yields in (37) negative definite, where ρi>0i=1,2,…,n is the diagonal entries of the positive definite diagonal matrix Λρ. It is worth noting that η˙ist indicates the velocity of each mobile sensing device.
Moreover, the choice of the velocity law for the mobile sensing device is decoupled, meaning that it is only taken into account for its own measurement. And in a real mobile sensor network, not only its velocity depends on itself, but also its neighbors contribute to its velocity law. Every mobile sensor (38), regardless of its velocity law, can be modified to(39)η˙ist=−ρiγiβ¯i∑j=1nWij1+κi∑k≠i∑j=1nWkj2,where κi>0,i,j,k=1,2,…,n.
Calculating the derivative of the second part of the Lyapunov function yields(40)∑i=1nL˙t,piηt+∑i=1nηt,piL˙t=2∑i=1npiLt,ALt≤−2∑i=1npia0Lt2≤−2∑i=1npia0α2Lt2.
By substituting (36), (39)–(40) into (32), we obtain(41)ℒVt≤∑i=1n−2+qi−1EAcηisteit2+∑i=1nqiγi2β¯i1−β¯iσc4−2pia0α2Lt2−2∑k≠iEekt−eit2−2∑i=1nρiγi2β¯i2∑j=1nWij21+κi∑k≠i∑j=1nWkj2.
The inequalities (26) are feasible, which implies ℒVt≤0.
Thus, it is easy to deduce from (41) and the embedding that(42)ℒVt≤−∑i=1nEAcηisteit2≤−α0∑i=1nEeit2,where α0>0 is a constant that contains the embedding constant α and the coercivity constant a0. By employing the Ito^ formula, combined with (42), we can obtain(43)EVt=EV0+∫0tℒVsds≤EV0−α0∫0t∑i=1nEeit2ds,which implies that(44)∫0t∑i=1nEeit2ds≤1α0V0.
In addition, it can confirm that∑i=1nEeit2 is uniformly continuous on 0,+∞. Consequently, it follows from Lemma 1 that(45)limt⟶+∞Eet2=0.
The proof is complete if the filtering error system (21) with wt=0 is globally asymptotically stable in the mean square, as stated in Definition 1.Remark 5.
In theory, if we select the Lyapunov functional as(46)Vt=−ert∑i=1neit,Acηisteit+ert∑i=1nLt,piLt,then the filtering error system (21) with wt=0 can also be proved to be globally exponentially stable in the mean square as the similar way in Theorem 1.


## 3.2.H∞ Consensus Performance Analysis

Next, we focus on analyzing theH∞ performance of the filtering error system (21) under the zero initial condition.Theorem 2.
The disturbance attenuation levelγ>0 and the filter parameter γi and Gi are given. Under the assumption (A1)–(A5), the zero solution of the filtering error system (21) with wt=0 is globally asymptotically stable in the mean square. The H∞ consensus performance (22) is achieved for all nonzero wt if, with the initial condition, such that the following matrix inequality holds:(47)Ψ=−α0+σb2dϱdϱ−γ2<0.Proof.
(42) implies Ψ<0, which is simple to prove. In consideration of Theorem 1, the filtering error system (21) is globally asymptotically stable in the mean square. Let us now concentrate on how the closed-loop system performs in terms of the H∞ consensus. Likewise, as in Theorem 1, we build the Lyapunov functional candidate Vt. The same line computation used in Theorem 1 results in(48)ℒVt≤−α0∑i=1nEeit2+2∑i=1nEDwt,−Acηisteit≤∑i=1nEζt,Ψ^ζt,where ζt=et,wtT and Ψ^=−α0dϱdϱ0.
In order to address theH∞ consensus performance of the system (21), we provide(49)J=E∑i=1n∫0Tz˜it2−γ2wt2dt=E∑i=1n∫0Tz˜it,z˜itdt−nγ2∫0Twt,wtdt.
For any nonzero external disturbanceswt from (47) to (48), under the zero initial condition, we obtain(50)J≤E∑i=1n∫0Tfz˜it,z˜it−γ2wt,wt+ℒVtdt≤E∑i=1n∫0Tfℬeit,ℬeit−γ2wt,wt+∑i=1nEζt,Ψ^ζtdt≤∑i=1n∫0TfEζt,Ψζtdt,where Ψ is defined in Theorem 2.
Along the same lines as in the argument of Theorem1, we can establish that J<0. Letting Tf⟶∞, we have(51)∑i=1n∫0Tz˜it,z˜itdt<nγ2∫0Twt,wtdt,namely,(52)1n∑i=1nz˜it2<γ2wt2.
The theorem’s proof here is completed.


## 4. Some Special Case Analysis

Owing to the generality of the result stated in Theorem1, we now discuss its few useful applications in the following section.The argument for following Theorem3 is omitted since it can be inferred from Theorem 1, Theorem 2, and the spatial distribution of moving sensors (8).Theorem 3.
The disturbance attenuation levelγ>0 and the filter parameter γi are given. Under assumptions (A1)–(A5) and inequalities (26), for the distributed parameter system (15), an H∞ consensus filter (20) is then constructed so that the filtering error system (21) with wt=0 is globally asymptotically stable in the mean square and also satisfies (22) under the zero initial condition for all non-zero wt if the mobile sensing device has a spatial distribution as (8), and its velocity law is as follows:(53)η˙ist=−ρiγiβ¯iWi1+κi∑j≠iWj2,where(54)Wi=∫ηis−εiηis+εi∂ciη∂ηciηei2t,ηdη+ci2ηis−εi+0ei2t,ηis−εi−ci2ηis+εi−0ei2t,ηis+εi,i,j=1,2,…,n,where ρi>0 denotes each sensor’s velocity gain, and the matrix inequality (47) holds.Hereafter, we discuss how to use a homogeneous mobile sensor network to solve theH∞ filtering issue for the error system (21).The spatial distribution of a homogeneous mobile sensor with a time-varying location function ofηist can be given by(55)cη;ηis=μ,ifη∈ηis−ε,ηis+ε,0,otherwise.As Theorem1 and Theorem 2 easily lead to this theorem, the proof is omitted.Theorem 4.
The disturbance attenuation levelγ>0 and the filter parameter γi are given. Under assumptions (A1)–(A5) and inequalities (26), for the distributed parameter system (15), an H∞ consensus filter (20) is then constructed so that the filtering error system (21) with wt=0 is globally asymptotically stable in the mean square and also satisfies (22) under the zero initial condition for all nonzero wt if the mobile sensing device has a spatial distribution as (55), its velocity law is as follows:(56)η˙ist=−ρiγiβ¯iμ2ei2t,ηis−ε−ei2t,ηis+ε1+κi∑j≠iej2t,ηjs−ε−ej2t,ηjs+ε2,i,j=1,2,…,n,where ρi>0 denotes each sensor’s velocity gain, and the matrix inequality (47) holds.Remark 6.
Actually, for an homogeneous mobile sensor network, by takingμ=cη, the location distribution of each moving sensor can be generally given as follows: (57)cη;ηis=cη,ifη∈ηis−ε,ηis+ε,0,otherwise.
In this case,cη is allowed to be constant or a piecewise smooth function in local.Remark 7.
In this paper, the case of data missing from the sensor network is taken into consideration for the distributedH∞ consensus filter for the distributed parameter system (1). Specially, let us consider that the measurement output (4) has no randomly occurred missing measurements, i.e., βit≡1∈R,i=1,2,…,n, and we choose μ=1, a common spatial distribution of a mobile sensor, as follows:(58)cη;ηis=1,ifη∈ηis−ε,ηis+ε,0,otherwise.
In the light of Theorem4, each moving sensor’s velocity (56) is able to be further simplified as(59)η˙ist=−ρiγiei2t,ηis−ε−ei2t,ηis+ε1+κi∑j≠iej2t,ηjs−ε−ej2t,ηjs+ε2,i=1,2,…,n,where ρi>0 is the velocity gain of each sensing device. This result is the same as the one in [8], which implies that the main results of this paper are extended to the earlier work.Remark 8.
Let us assume that the homogeneous sensor network has all of the sensors fixed. In other words,ηist≡ηis0:=η¯is is a constant that is independent of time. In this way, the time-spatial operator of the measurement output operator Cηst is changed to the spatial operator Cη¯is. Since Cη¯is is a constant operator, we can also express Cη¯is:=C. The measurement output equation (4) can be rewritten as(60)yt=ΛβtCLt.
For theith stationary sensor, the suitable filter is(61)L^˙it=AL^it+C∗γiyit−β¯iCL^it−Gi∑j≠iL^it−L^jt,z^it=ℬL^it,where L^i0=L^i0≠L0 for all i=1,2,…,n. As a result, (15) and (61) can be used to establish the filtering error system in the lines as follows:(62)e˙it=Aceit−C∗γiβit−β¯iCLt+Gi∑j≠iejt−eit+Dwt,z˜it=ℬeit,where Ac=A−γiβ¯iC∗C and ei0=L0−L^i0≠0. The following corollary is easily shown in Theorem 3.Corollary 1.
The disturbance attenuation levelγ>0 and the filter parameter γi and Gi are given. For the distributed parameter system (15), an H∞ consensus filter (61) is then constructed so that the filtering error system (62) with wt=0 is globally asymptotically stable in the mean square and also satisfies (22) for all nonzero wt under the zero initial condition if assumptions (A1)–(A5), inequality (26), and the matrix inequality (47) hold.


## 5. Numerical Examples

To illustrate the usefulness of the distributedH∞ consensus filter created in this work, we give a simulated example in this section. The spatial distribution process of temperature in a chemical reactor having Dirichlet boundary conditions and initial conditions is L0,η=sinπηe−9η2,η in [0, 1]. The evolution of the system is illustrated in Figure 1. The diffusion coefficient is a0=0.004. The bounded function is φLt,η=1.2sin0.6Lt,η. In a kind of spatially distributed process, three mobile sensing devices are considered to collect data and η1s0=0.15,η2s0=0.5, and η3s0=0.85 are selected for their initial positions.Figure 1
The state evolution of parabolic distributed parameter processes.The spatial distribution of each moving sensing device inηist is described by the time-varying location function, which is provided by(63)cη;ηis=1,ifη∈ηis−0.07,ηis+0.07,0,otherwise.The probabilities are considered to beβ¯1=0.9,β¯2=0.85, and β¯3=0.8. As shown in Figure 2, the measurement with missing random data is observed by moving sensors.Figure 2
The output of moving sensors with random missing data.Initial conditions are assumed to beL^10,η=L^20,η=L^30,η=0 for the distributed H∞ consensus filter. The filter gains are given by γ1=80,γ2=85, and γ3=90. Three distributed H∞ consensus filters are employed to illustrate the evolution of the filtering error system in Figures 3–5. The filtering error z˜it is given in Figure 6. The output zt and associated estimates for mobile sensors are shown in Figure 7. The output zt of the ith filter i=1,2,3 in the moving sensing device and its estimated value are shown in Figure 7.Figure 3
Filter error system development for filter 1.Figure 4
Filter error system development for filter 2.Figure 5
Filter error system development for filter 3.Figure 6
Output of plant and filters.Figure 7
Output estimation errors of filters.Three fixed-in-space sensors are considered to be a comparison, which are located atη1s=0.15,η2s=0.50, and η3s=0.85. The trajectories of the three sensors in stationary and moving scenarios are illustrated in Figure 8.Figure 8
The trajectories of three mobile sensors.


## 6. Conclusions

TheH∞ distributed consensus filtering issue has been investigated in this paper for an array of parabolic distributed parameter systems with multiple missing measurements. An effective Lyapunov direct approach, which ensures a specific level of H∞ consensus disturbance rejection attenuation, has been proposed to build the filtering error system that is globally asymptotically stable in the mean square for all permissible randomly occurring missing data. Several sufficient conditions can be obtained within the optimized framework, including the velocity law of the moving sensor, which can ensure a faster convergence of the filtering error to zero. Finally, a numerical simulation is adopted to demonstrate the usefulness of the obtained results of the study.



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*Source: 9740248-2023-02-13.xml*