# Adaptive Finite-Horizon Group Estimation for Networked Navigation Systems with Remote Sensing Complementary Observations under Mixed LOS/NLOS Environments

Authors: Chao Gao; Jianhua Lu; Guorong Zhao; Shuang Pan
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

Networked navigation system (NNS) enables a wealth of new applications where real-time estimation is essential. In this paper, an adaptive horizon estimator has been addressed to solve the navigational state estimation problem of NNS with the features of remote sensing complementary observations (RSOs) and mixed LOS/NLOS environments. In our approach, it is assumed that RSOs are the essential observations of the local processor but suffer from random transmission delay; a jump Markov system has been modeled with the switching parameters corresponding to LOS/NLOS errors. An adaptive finite-horizon group estimator (AFGE) has been proposed, where the horizon size can be adjusted in real time according to stochastic parameters and random delays. First, a delay-aware FIR (DFIR) estimator has been derived with observation reorganization and complementary fusion strategies. Second, an adaptive horizon group (AHG) policy has been proposed to manage the horizon size. The AFGE algorithm is thus realized by combining AHG policy and DFIR estimator. It is shown by a numerical example that the proposed AFGE has a more robust performance than the FIR estimator using constant optimal horizon size.


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## Body

## 1. Introduction

Networked navigation system (NNS) [1, 2], which determines its navigational state individually from the cooperative information taken with respect to spatial neighbors, has been used for a variety of purposes, such as cooperative localization [3], multivehicle cooperative guidance [4], and spatially distributed estimation [5]. NNS typically takes advantage of state estimations for accurate localization essentially based on the remote sensing observations (RSOs), which is transmitted by the cooperative NNSs through a wireless medium. Thus, during the design of navigational state estimator, several problematic issues in NNS should be considered, induced by the observation features and the transmission environment. In this research, the impact of mixed LOS/NLOS environment and complementary feature of RSOs will be investigated.Remote sensing observations (RSOs), such as image, relative measurement, and sink knowledge, always have complementary feature, which means every observed data just comprises partial of system dynamics; thus one useful observation of estimator can only be obtained by the integration of multiple synchronous RSOs. Note that these RSOs are ubiquitous in large-scale systems [6], especially in wireless networked system (WNS) [7], and thus the state estimation based on the sensors of RSO data is brought forward. In [8], a particle filter based track-before-detect (TkBD) scheme was proposed for multiple sensors, where the sensors were asynchronous and heterogeneous; however, the TkBD schemes are computationally expensive and are mainly applicable in scene of low signal-to-noise ratios. Earlier research of RSO focused on optical sensors [9], in which the sensor noise is assumed to be Gaussian. More recent efforts focus on the radar problem, where the pixel intensity is either Rayleigh or Rician distributed depending on the absence or presence of objectives [10]. Although the extant literature considers various types of RSO sensors, the researches on the complementary property of wireless networks are quite few, let alone the packet-based RSO of NNS, that is, the navigational knowledge of the sender and the relative measurements.Non-line-of-sight (NLOS) propagation might take place in a wireless network whenever there are obstacles hindering the radio signal path between the objective and its cooperative neighbors, especially in an urban region or the indoor environment. Whenever measured by received signal strength (RSS) or angle of arrival (AOA), NLOS propagation often induces transmission delay or dropout of the packets, and thus large localization errors might occur if not addressed [11]. Several strategies have been proposed for mitigating the bad effect of NLOS errors, such as statistical identification [12], hypothesis test [13], and robust parameter estimation [14]. In [15], the switching behavior of NLOS and the line-of-sight (LOS) has been described by a hidden Markov Model, and an estimator has been designed by combining Kalman filter with the interacting multiple model (IMM) approach. These results are further extended to nonlinear AOA measurement by using the MUSIC method in [16]. Similar approach has also been used to observe a mobile objective in mixed LOS/NLOS environments by using individual measurement and LOS detection [17]. However, little research has been investigated to mitigate the delay and dropout problems caused by NLOS propagation. In [18], a joint AOA and delay estimation method has been proposed for space-time coherent distributed signals based on search. The problem of joint AOA/RSS based state estimation for NNS in mixed LOS and NLOS environment has not been studied.In reality, almost all networked systems can record the results over a recent finite horizon; for such time-varying systems, one would be more interested in the optimal estimation based on the finite recent data. When it comes to the filter design issue, finite-horizon estimation is an effective approach and gives rise to many available methods in terms of moving horizon estimation [1, 19], finite-horizon H∞ fault estimation [20], and finite-horizon FIR estimation [21]. However, finite-horizon estimation approach has its own limitations, especially on expensively computational burden [22], with the reasons in terms of the adoption of finite recent measurements, the inversion computation [23], and the singularity problem [24]. One of the improvement methods is to manage the horizon size (window length), and several approaches [25] have been proposed. Although the existing approaches are effective with linear time-invariant systems, they can be less effective with wireless networked systems, with the reason that the constant (i.e., fixed) horizon size of the existing methods may be not able to handle the network-induced constraints owing to mixed LOS/NLOS environment. Thus, it is desirable to adapt the horizon size in finite-horizon estimation, and this motivates us to investigate this issue.Summarizing the above analysis, it can be concluded that there is a great need to examine how the mixed LOS/NLOS environments affect the performance of the distributed estimation over a sensor network with RSO data. To address the effects of NLOS errors and random delays, a jump Markov system has been modeled with the switching parameters corresponding to LOS/NLOS errors. An adaptive finite-horizon group estimator (AFGE) has been proposed, where the horizon size can be adjusted in real time according to stochastic parameters and random delays. A numerical example is proposed to verify the effectiveness of the proposed AFGE scheme. The main contributions of this research can be highlighted as follows. (i) The observation model of NNS is a special one that has complementary property and is affected by switching LOS/NLOS errors and random transmission delay. (ii) An observation reorganization scheme and a complementary fusion strategy have been introduced to DFIR estimation procedure to mitigate the impacts of delay and complementary property, respectively. (iii) The proposed AFGE scheme contains an AHG policy, which is proposed to manage the horizon size.The remainder of this paper is organized as follows. In Section2, the graph-based network background is introduced, while the NNS model and some preliminaries are briefly outlined. Section 3 describes the derivation of a new DFIR estimator. Section 4 proposes the AHG method and completes the AFGE. In Section 5, an application of clustered UAVs is given for the designed AFGE. Finally, the conclusions are drawn in Section 6.Notation.  uG stands for the Euclidean norm of a generic vector u; uG2=uTGu, where G is a symmetric positive semidefinite matrix. Given a generic vector ut, ut-Nt=ut-N,t≜col(ut-N,ut-N+1,…,ut); a generic matrix At-Nt=At-N,t≜col(At-N,At-N+1,…,At), t=N,N+1,…; L{y1,…,yn} denotes the linear space spanned by y1,…,yn.


## 2. Preliminaries and Problem Formulation

In this section, we introduce some preliminaries related to the description of communication network and then present the problem setup.


### 2.1. Graph-Based Network Description

Suppose thatNΣ vehicles form into an Ad hoc network, where every vehicle acts as a networked node and carries out the uniform communication protocol, that is, [1, 2, 19]. The communication topology of information flow between networked vehicles is described by a directed graph G=(Cg,Eg,Rg), where Cg∈{1,…,NΣ} is the finite set of networked vehicles, Eg⊆Cg×Cg is the set of edges, Rg=(ri)∈Rn is the set of action radius, and i∈Cg represents the sender. The edge e[i→j]=(i,j)∈Eg indicates that node j can receive the information from node i; that is, (i,j)∈Eg⇒d[i→j]≤ri. Specifically, we define the set of neighbors as follows.Definition 1.
Letxp[i]∈R3 be the position of node ƛ∈Cg; the neighbor set of node ƛ at time t can be defined as(1)Ntƛ=j∈Cg:rinf≤xpjt-xpƛt≤rsup,ƛ≠j,Cg=1,2,…,NΣ,where rsup and rinf denote thesupremum andinfimum radii of available communication range, respectively. Note that the relative distances between node ƛ and Nt[ƛ] are changeable with time t; thus the member of Nt[ƛ] may also be changed dynamically. Besides, the effective area for cooperative neighbors can be defined by rsup and rinf, namely, receive-search area (RSA), as depicted in Figure 1.Figure 1
Remote sensing neighbors for networked navigation system.The model of RSA is defined as follows.Definition 2.
Given a generic nodeƛ∈Cg, its geographic neighbor j∈Cg is defined as in the RSA of node ƛ, if node j is in the hollow sphere between rinf and rsup, which are determined by RSS method as(2)lg⁡rsup=PLdB-γsup10λ-3.244λ-lg⁡fMHz,lg⁡rinf=PLdB-γinf10λ-3.244λ-lg⁡fMHz,where γsup and γinf denote the received SNRs corresponding to rsup and rinf, respectively, PLdB is the propagation loss (dB) in free space propagation model [12], λ is the path loss exponent, 2≤λ≤4, and fMHz is the electromagnetic frequency.


### 2.2. NNS Model and Problem Formulation

In this paper, the NNS is assumed as the essential navigation equipment of a generic vehicleƛ∈Cg. The dynamics of node S can be modeled as the discrete-time linear stochastic system(3)xt+1=Φtxt+Γtwt, where xt denotes system state; Φt and Γt denote the system matrices which can be used to represent the motions such as the velocity model and coordinate turn model. The process noise wt is zero-mean white Gaussian with covariance matrix Qt.A set of cooperative UAVs that provide two kinds of observations, (i) navigational states of the observed neighbors, denoted byxt{N}={x~t[i]x~t[i],i∈Nt[ƛ]}, x~t[i]=(x~x,t[i],x~y,t[i],x~z,t[i]), and (ii) relative measurements (RM) between ƛ and the neighbors i∈Nt[ƛ], denoted by zt{N}={zt[i]zt[i],i∈Nt[ƛ]}, including relative distance and relative angle, are assumed to be calculated by RSS and AOA, respectively. The detailed deductions can be found in [12, 16], with the basic equations as(4)zRSSi=α-10λlg⁡x~i-x^ƛ+ηshadi,zAOA,θi=arctan⁡x~yi-x^yƛx~xi-x^xƛ+ηAOA,θi,zAOA,ϕi=arctan⁡x~zi-x^zƛx~x,yi-x^x,yƛ+ηAOA,ϕi,i=1,2,…,l.Compared to the previous DSE problems [5, 20, 25], a typical feature here is that the observations in xt{N} and zt{N} are mutually complementary; that is, each local observation of node ƛ can only be derived through the fusion of multiple xt[i] and zt[i](i∈Nt[ƛ]). The measure noise covariance Rs[i] in Cartesian coordinates is(5)Rsi=ϕzRSSi,zAOA,θi,zAOA,ϕi,ηshadi,ηAOA,θi,ηAOA,ϕi,where zRSS[i], zAOA,θ[i], zAOA,ϕ[i], ηshad[i], ηAOA,θ[i], and ηAOA,ϕ[i] are defined in (4). In LOS environment, the measurements (4) are corrupted by the noise εt[i], which can be modeled as zero-mean white Gaussian noise; that is, εt[i]~N(εt[i];0,Rt,1[i]). In NLOS environment, the measurement is corrupted by two kinds of error sources: Gaussian noise εt[i] and measurement bias error et[i]. The bias error can be modeled as a biased distribution; that is, et[i]~N(et[i];μt[i],Σt[i]). It is also assumed that εt[i] and et[i] are independent. Thus, the CM noise in (4) can be represented by(6)uti=εti,LOS  condition,εti+eti,NLOS  condition.For the sake of notations, the measurement equation (4) can be represented in a uniform form:(7)fxtN,ztN≜zti=ϕtixt+utiγti,i=1,2,…,l, where ϕt[i] is the nonlinear measurement function, depending on specific RM methods; the measurement noise ut[i](γt[i]) is assumed to be zero-mean white Gaussian with covariance matrix Rt,1[i] in the LOS condition; that is, ut,1[i]∈N(ut[i];0,Rt,1[i]); and it is assumed to be white Gaussian with mean μt[i] and covariance matrix Rt,2[i] in the NLOS condition; that is, ut,2[i]∈N(ut[i];μt[i],Rt,2[i]). γt[i] is introduced to indicate the transition of LOS and NLOS measurement errors; that is, γt[i]=1 represents the LOS condition and γt[i]=2 represents the NLOS condition. The TPM of γt[i] is assumed to be Π(2)=[πpq(2)]2×2 with πpq(2)=Pr⁡γti=q∣γt-1i=p. It should be pointed out that the two-state Markov chain γt[i] is dependent on the cooperative UAV, since the objective might be in LOS condition for one UAV but in NLOS condition for another one.Under the above mixed LOS/NLOS propagation environment, transmission delays (TDs) may decay the accuracy of relative measurement. To mitigate this disadvantage, the observed CM with TDs is modeled as(8)yti=∑τ=0ħκt,τizt-τi=κt,τiϕt-τixt-τ+κt,τiut-τiγt-τi,i=1,2,…,l with κt,τ[i] defined as a binary random variable indicating the arrival of the observation packet for state x(t-τ); that is,(9)κt,τi=1,ifzt-τiis  received  at  the  instantt;0,otherwise;i∈1,2,…,l.Assumption 3.
Random delayτ(t) is bounded with 0<τ(t)≤ħ, where ħ is given as the length of memory buffer, and its probability distributions are Prob(τ(t)=τ)=πτ, τ=1,…,ħ. If the CMs transmitted to the objective with a delay larger than ħ, they will be considered as the ones lost completely. Thus, the property 0≤∑τ=1ħπτ≤1 is satisfied. Also τ(t) is independent of x(0), w(0), and u(0).Obviously,Prob(κt,τ[i]=1)=πτ, τ=1,…,ħ. Under Assumption 3, we know that the possible received observations at time t (t≥ħ) are (10)y~ti=κt,0iztiTκt,1izt-1iT⋯κt,ħizt-ħiTT.In the real-time networked systems, the statex(t) can only be observed at most one time, and thus κt,τ[i], τ=1,…,ħ, must satisfy the following property:(11)κt+j,ji×κt+k,ki=0,j≠k.Whent<ħ, the observation of (10) is written as(12)y~ti=κt,0iztiT⋯κt,tiztiT0⋯0T,where we set y~t[i]≡0 for t<τ≤ħ. Then the estimation problems considered in this paper can be stated as follows.Problem E t. For a generic node ƛ∈Cg, given the set of navigation knowledge from the neighbors x~s[i] (i∈Nt[ƛ]) with the stamp s (s>0), the corresponding relative measures y~k[i] (k∈[t-Ξ[i],t]) subject to NLOS errors and random delay, with the information κk,τ[i], find a linear minimum mean square error filter x^ƛ(t∣t) for navigational state of the target, x(t), such that(13)Jƛ=Ext-x^ƛt∣t∣x^sis≥0;y~kik=t-Ξit;κk,τik=t-Ξit,i∈Ntƛ=1,2,…,lis minimized, while the filter gain is stochastic.


## 2.1. Graph-Based Network Description

Suppose thatNΣ vehicles form into an Ad hoc network, where every vehicle acts as a networked node and carries out the uniform communication protocol, that is, [1, 2, 19]. The communication topology of information flow between networked vehicles is described by a directed graph G=(Cg,Eg,Rg), where Cg∈{1,…,NΣ} is the finite set of networked vehicles, Eg⊆Cg×Cg is the set of edges, Rg=(ri)∈Rn is the set of action radius, and i∈Cg represents the sender. The edge e[i→j]=(i,j)∈Eg indicates that node j can receive the information from node i; that is, (i,j)∈Eg⇒d[i→j]≤ri. Specifically, we define the set of neighbors as follows.Definition 1.
Letxp[i]∈R3 be the position of node ƛ∈Cg; the neighbor set of node ƛ at time t can be defined as(1)Ntƛ=j∈Cg:rinf≤xpjt-xpƛt≤rsup,ƛ≠j,Cg=1,2,…,NΣ,where rsup and rinf denote thesupremum andinfimum radii of available communication range, respectively. Note that the relative distances between node ƛ and Nt[ƛ] are changeable with time t; thus the member of Nt[ƛ] may also be changed dynamically. Besides, the effective area for cooperative neighbors can be defined by rsup and rinf, namely, receive-search area (RSA), as depicted in Figure 1.Figure 1
Remote sensing neighbors for networked navigation system.The model of RSA is defined as follows.Definition 2.
Given a generic nodeƛ∈Cg, its geographic neighbor j∈Cg is defined as in the RSA of node ƛ, if node j is in the hollow sphere between rinf and rsup, which are determined by RSS method as(2)lg⁡rsup=PLdB-γsup10λ-3.244λ-lg⁡fMHz,lg⁡rinf=PLdB-γinf10λ-3.244λ-lg⁡fMHz,where γsup and γinf denote the received SNRs corresponding to rsup and rinf, respectively, PLdB is the propagation loss (dB) in free space propagation model [12], λ is the path loss exponent, 2≤λ≤4, and fMHz is the electromagnetic frequency.


## 2.2. NNS Model and Problem Formulation

In this paper, the NNS is assumed as the essential navigation equipment of a generic vehicleƛ∈Cg. The dynamics of node S can be modeled as the discrete-time linear stochastic system(3)xt+1=Φtxt+Γtwt, where xt denotes system state; Φt and Γt denote the system matrices which can be used to represent the motions such as the velocity model and coordinate turn model. The process noise wt is zero-mean white Gaussian with covariance matrix Qt.A set of cooperative UAVs that provide two kinds of observations, (i) navigational states of the observed neighbors, denoted byxt{N}={x~t[i]x~t[i],i∈Nt[ƛ]}, x~t[i]=(x~x,t[i],x~y,t[i],x~z,t[i]), and (ii) relative measurements (RM) between ƛ and the neighbors i∈Nt[ƛ], denoted by zt{N}={zt[i]zt[i],i∈Nt[ƛ]}, including relative distance and relative angle, are assumed to be calculated by RSS and AOA, respectively. The detailed deductions can be found in [12, 16], with the basic equations as(4)zRSSi=α-10λlg⁡x~i-x^ƛ+ηshadi,zAOA,θi=arctan⁡x~yi-x^yƛx~xi-x^xƛ+ηAOA,θi,zAOA,ϕi=arctan⁡x~zi-x^zƛx~x,yi-x^x,yƛ+ηAOA,ϕi,i=1,2,…,l.Compared to the previous DSE problems [5, 20, 25], a typical feature here is that the observations in xt{N} and zt{N} are mutually complementary; that is, each local observation of node ƛ can only be derived through the fusion of multiple xt[i] and zt[i](i∈Nt[ƛ]). The measure noise covariance Rs[i] in Cartesian coordinates is(5)Rsi=ϕzRSSi,zAOA,θi,zAOA,ϕi,ηshadi,ηAOA,θi,ηAOA,ϕi,where zRSS[i], zAOA,θ[i], zAOA,ϕ[i], ηshad[i], ηAOA,θ[i], and ηAOA,ϕ[i] are defined in (4). In LOS environment, the measurements (4) are corrupted by the noise εt[i], which can be modeled as zero-mean white Gaussian noise; that is, εt[i]~N(εt[i];0,Rt,1[i]). In NLOS environment, the measurement is corrupted by two kinds of error sources: Gaussian noise εt[i] and measurement bias error et[i]. The bias error can be modeled as a biased distribution; that is, et[i]~N(et[i];μt[i],Σt[i]). It is also assumed that εt[i] and et[i] are independent. Thus, the CM noise in (4) can be represented by(6)uti=εti,LOS  condition,εti+eti,NLOS  condition.For the sake of notations, the measurement equation (4) can be represented in a uniform form:(7)fxtN,ztN≜zti=ϕtixt+utiγti,i=1,2,…,l, where ϕt[i] is the nonlinear measurement function, depending on specific RM methods; the measurement noise ut[i](γt[i]) is assumed to be zero-mean white Gaussian with covariance matrix Rt,1[i] in the LOS condition; that is, ut,1[i]∈N(ut[i];0,Rt,1[i]); and it is assumed to be white Gaussian with mean μt[i] and covariance matrix Rt,2[i] in the NLOS condition; that is, ut,2[i]∈N(ut[i];μt[i],Rt,2[i]). γt[i] is introduced to indicate the transition of LOS and NLOS measurement errors; that is, γt[i]=1 represents the LOS condition and γt[i]=2 represents the NLOS condition. The TPM of γt[i] is assumed to be Π(2)=[πpq(2)]2×2 with πpq(2)=Pr⁡γti=q∣γt-1i=p. It should be pointed out that the two-state Markov chain γt[i] is dependent on the cooperative UAV, since the objective might be in LOS condition for one UAV but in NLOS condition for another one.Under the above mixed LOS/NLOS propagation environment, transmission delays (TDs) may decay the accuracy of relative measurement. To mitigate this disadvantage, the observed CM with TDs is modeled as(8)yti=∑τ=0ħκt,τizt-τi=κt,τiϕt-τixt-τ+κt,τiut-τiγt-τi,i=1,2,…,l with κt,τ[i] defined as a binary random variable indicating the arrival of the observation packet for state x(t-τ); that is,(9)κt,τi=1,ifzt-τiis  received  at  the  instantt;0,otherwise;i∈1,2,…,l.Assumption 3.
Random delayτ(t) is bounded with 0<τ(t)≤ħ, where ħ is given as the length of memory buffer, and its probability distributions are Prob(τ(t)=τ)=πτ, τ=1,…,ħ. If the CMs transmitted to the objective with a delay larger than ħ, they will be considered as the ones lost completely. Thus, the property 0≤∑τ=1ħπτ≤1 is satisfied. Also τ(t) is independent of x(0), w(0), and u(0).Obviously,Prob(κt,τ[i]=1)=πτ, τ=1,…,ħ. Under Assumption 3, we know that the possible received observations at time t (t≥ħ) are (10)y~ti=κt,0iztiTκt,1izt-1iT⋯κt,ħizt-ħiTT.In the real-time networked systems, the statex(t) can only be observed at most one time, and thus κt,τ[i], τ=1,…,ħ, must satisfy the following property:(11)κt+j,ji×κt+k,ki=0,j≠k.Whent<ħ, the observation of (10) is written as(12)y~ti=κt,0iztiT⋯κt,tiztiT0⋯0T,where we set y~t[i]≡0 for t<τ≤ħ. Then the estimation problems considered in this paper can be stated as follows.Problem E t. For a generic node ƛ∈Cg, given the set of navigation knowledge from the neighbors x~s[i] (i∈Nt[ƛ]) with the stamp s (s>0), the corresponding relative measures y~k[i] (k∈[t-Ξ[i],t]) subject to NLOS errors and random delay, with the information κk,τ[i], find a linear minimum mean square error filter x^ƛ(t∣t) for navigational state of the target, x(t), such that(13)Jƛ=Ext-x^ƛt∣t∣x^sis≥0;y~kik=t-Ξit;κk,τik=t-Ξit,i∈Ntƛ=1,2,…,lis minimized, while the filter gain is stochastic.


## 3. Delay-Aware FIR Estimator with Delayed Complementary Observations

In this section, we aim to establish a unified framework to solve the addressed optimal state estimation problem in the simultaneous presence of network-induced delay and complementary property under the mixed LOS/NLOS environment, while the observations are decoded from the packets ofl spatially distributed nodes.


### 3.1. Delayed Observation Reorganization

In our research, the remote sensing observations (i.e.,{zt[i],xt[i]}) are impacted by delay and complementary issues and thus should not be adopted directly by the estimators. In this subsection, we focus on the delay issue; more specifically, we propose a preestimating scheme, including two steps, to reorganize the delay observations without dimensional augment.Step 1.
We rearrange the complementary observations{y[i](k)t-Ξ[i]t,i∈Nt[ƛ]}, which are received in horizon [t-Ξ[ƛ],t], that is, (8), and rewrite them into two parts with the batch form as(14)y~ħs≜y0jTsj=1l,yħjTs+ħj=1lT,t-Ξs-ħ≤s≤t-ħ,y~t-ss≜y0jTsj=1l,yt-sjTtj=1lT,t-ħ<s≤t.
Then, reorganized observationy~(·) is delay-free and satisfies(15)y~ħs≜H~ħsxs+v~ħs,γs,s+ħi,t-Ξs-ħ≤s≤t-ħ,y~t-ss≜H~t-ssxs+v~t-ss,γs,ti,t-ħ<s≤t,where(16)H~ħs=κs,0iϕsii=1lTκs+ħ,ħiϕs+hii=1lTT,H~t-ss=κs,0iϕsii=1lTκt,t-siϕtii=1lTT,v~ħs,γs,s+ħi=κs,0iusiγsii=1lTκs+ħ,ħius+ħiγs+ħii=1lTT,v~t-ss,γs,ti=κs,0iusiγsii=1lTκt,t-siutiγtii=1lTT,with vħ(s) and vt-s(s) being white noises of zero means and covariance matrices:(17)R~ħs=diag⁡κs,0iRii=1l,κs+ħ,ħiRii=1l,R~t-ss=diag⁡κs,0iRii=1l,κs+ħ,ħiRii=1l.Moreover, we can follow the result in [1] that the following lemma is true.Lemma 4.
For the given time horizon[t-Ξ[ƛ],t], the linear space L1≜{y(k)}k=t-Ξ[ƛ]t generated by {y(k)}k=t-Ξ[ƛ]t is equivalent to the linear space of(18)L2≜y~ħss=t-Ξƛ-ħt-ħ-1;y~t-sss=t-ħt, which implies that L2 contains the same information as L1.Step 2.
We further fuse the complementary observations (15) of the same stamp s∈[t-Ξ[ƛ]-ħ,t) into one complete measurement y~m[ƛ](s) (m∈{ħ,t-s}) prior to the local estimation process, and thus(19)y~mƛs=R-·∑i=1lgy~mis,xmis,m∈h,t-s,s=t-Ξƛ-ħ,…,t, where R- denotes the weighted matrix; g(y~m[i](s),xm[i](s)) is the relative measure function. Thus, the local measurements y~m[ƛ](s)∈Rny in (19) can be rewritten in the reorganized linear space L2 as(20)y~mƛs=ϑsƛH~mƛsxs+v~mƛs,γmƛ, where ϑs[ƛ] represents the failed fusion of y~[ƛ](s) at time stamp s=t-Ξ[ƛ]-ħ,…,t-1, which takes the value of 0 or 1, and is modeled as Bernoulli distributed white sequences satisfying Probϑs[ƛ]=1=ϑ-[ƛ]; 0<ϑ-[ƛ]≤1.Remark 5.
Note thatϑs[ƛ] (s=t-Ξ[ƛ]-ħ,…,t-1) is introduced to represent the failure of the preestimating algorithm (19) at time s, with the reason that the numbers of reorganized CMs y~ħ∣t(s) or y~t-s∣t(s) are less than a lower limit [9, 12]. Thus, through preestimating transformation, the remote sensing CMs in (14) with high dimension are transformed into the data missing in (20) with lower dimension.
Besides,R- and g(y~m[i](s),xm[i](s)) in (19) will be deduced through a complementary fusion strategy, with the details as follows.


### 3.2. Complementary Fusion Strategy

Let{y~m[i](s),i=1,2,…,l} be the set of complementary observations generated at instant s, the relative measures between objective ƛ and its neighbors i∈Ns[ƛ], that is, relative distance and relative angle, are calculated through RSS and AOA, respectively, as depicted in (4), and the measure noise covariance R~m[i](s) in (17).Hereafter, the fused measurement covarianceR- and the fused measurement variable ym[ƛ](s) (m∈{ħ,t-s}) can be expressed as(21)R-=∑i=1lR~mis-1-1,m∈ħ,t-s(22)ymƛs=R-·∑i=1lR~mis-1y~mis+xmls-xmis.Besides, the sensitivity matrix of the fused measurement variable (22) can be written as(23)H-=R-·∑i=1lR~mis-1Hi;hereinto, H[i] is the sensitivity matrix corresponding to y~m[i](s):(24)Hi=-I3×303×3.Note that the estimation error in this research may not always fall within the confidence level of the predicted covariance boundary. To avoid this problematic scenario, we introduce a new weighting factorψi,y as follows.Definition 6.
Given the CMsy~m[i](s), m∈{ħ,t-s}, sensitivity matrix H[i], and the inverse measurement noise variance, (R~m[i](s))-1, the weighting factor, ψi,y, is applied before they are fused; more specifically, it is defined as(25)ψi,y=ϱi2ϱTϱ,where ∑i=1lψi,y=1; ϱ=ϱ1ϱ2⋯ϱlT.The weighting factorψi,y is a function of (4). For instance, RSS vector is defined as Pm,rss=y~m,rss1⋯y~m,rsslT, m∈{ħ,t-s}. In the proposed weight strategy, the weighting factor of each CM is assigned based on the relative measures between i∈Ns[ƛ] and ƛ. Lower weight is assigned to the neighbors i∈Ns[ƛ] closer to ƛ. In other words, weighting factor ψi,y is selected to be proportional to the relative measures between the neighbors. The last measure y~m,rss[l], which is equivalent to max(Pm,rss), has highest ψi,y. Define the terms ΔPm,rss and ϱ as(26)ΔPm,rssi=y~m,rssl-y~m,rssi,m∈ħ,t-s,ϱi=1-ΔPm,rssiΔPm,rssi,m∈ħ,t-s,where ΔPm,rss=[ΔPm,rss[1],ΔPm,rss[2],…,ΔPm,rss[l]].Thus, the newly fused measurement noise covariance in (21) can be rewritten as(27)R-=l·∑i=1lψi,yR~mis-1-1,m∈ħ,t-s.Furthermore, the sensitivity matrix in (23) can be rewritten as(28)H-=∑i=1lψi,yR~mis-1-1∑i=1lψi,yR~mis-1Hi,m∈ħ,t-s.Meanwhile, the newly fused measurement vector in (22) can be rewritten as(29)ymƛs=R-·∑i=1lgy~mis,xmis=l·∑i=1lψi,yR~mis-1-1∑i=1l1l·ψi,yR~mis-1y~mis+xmls-xmis,m∈ħ,t-s,where(30)gy~mis,xmis=1l·ψi,yR~mis-1y~mis+xmls-xmis,m∈ħ,t-s.Note that the resulting problem is solved by a robust recursive scheme, which allows one to determine approximate estimates with a reduced computational condition. The details are presented in the following subsection.


### 3.3. Delay-Aware FIR Estimation Algorithm

In this subsection, we proposed a novel unbiased FIR estimator, namely, delay-aware FIR (DFIR) estimation algorithm, to realize the network-based state estimation of a generic nodeƛ∈Cg based on the cooperative observation of its neighborhood Ns[ƛ]. Note that several unique network-induced constraints are involved, including complementary observations, Markov jump parameters, and random delay; hereinto, partial constraints have been transformed in Sections 3.1 and 3.2, which may simplify the design procedure of DFIR.Prior to the main result of this section, let us recall a useful lemma that will be useful in DFIR algorithm.Lemma 7 (see [24]).
Suppose that the general trace optimization (GTO) problem is given as(31)minH-tr⁡H-A-BCH-A-BT+H-DH-T,subject  toH-E=F,where A, B, C, D, E, and F denote the constant matrices with appropriate dimensions: C=CT>0 and D=DT>0. tr(J) is the trace of the main diagonal of J.Thus, the unique solution to the GTO problem can be obtained as(32)H-=FBETΛ-1E-1ETΛ-1CATΛ-1I-EETΛ-1E-1ETΛ-1,where Λ=ACAT+D.On the recent finite-time horizon[t-Ξ[ƛ]-ħ,t-1] of reorganized linear space L2, the receding horizon filter for (3) and (20) can be written as (33)x^tt-1ƛ=∑k=t-Ξƛ-ħt-1H~t-k,γsy~ħƛs=H~γsY~t-1ƛ, where H~γ(s) is the gain matrix, Y~t-1[ƛ] is the augmented measurement vector, and γ(s) is the set of switching factors on the horizon [t-Ξ[ƛ]-ħ,t-1] defined by (34)–(36), respectively, as(34)H~γs≜H~Ξƛ+ħ,γsH~Ξƛ+ħ-1,γs⋯H~1,γs,(35)Y~t-1ƛ≜y~t-Ξƛ-ħƛTy~t-Ξƛ-ħ+1ƛT⋯y~t-1ƛTT,(36)γs≜γt-Ξƛ-ħƛ,γt-Ξƛ-ħ+1ƛ,…,γt-1ƛ.For ease of notation only, we express the horizon size of estimatorΞ[ƛ]+ħ as Ξ~; that is, Ξ~≜Ξ[ƛ]+ħ, Y~t-1[ƛ] can be rewritten in the batch form as(37)Y~t-1ƛ=FΞ~,γsxt-Ξ~+HΞ~,γsWt-1+Vt-1,γs, where(38)FΞ~,γs≜H~t-Ξ~,γt-Ξ~ƛϜt-Ξ~t-1⋮H~t-Ξ~+1,γt-Ξ~+1ƛϜt-2t-1H~t-1,γt-1ƛΦt-1-1,Wt-1≜wt-Ξ~wt-Ξ~+1⋮wt-1,Vt-1,γs≜v~t-Ξ~,γt-Ξ~ƛv~t-Ξ~+1,γt-Ξ~+1ƛ⋮v~t-1,γt-1ƛ,HΞ~,γs≜H~t-Ξ~,γt-Ξ~ƛΦt-Ξ~-1Γt-Ξ~H~t-Ξ~,γt-Ξ~ƛϜt-Ξ~t-Ξ~+1Γt-Ξ~⋯H~t-Ξ~,γt-Ξ~ƛϜt-Ξ~t-1Γt-Ξ~0H~t-Ξ~+1,γt-Ξ~+1ƛΦt-Ξ~+1-1Γt-Ξ~+1⋯H~t-Ξ~+1,γt-Ξ~+1ƛϜt-Ξ~+1t-1Γt-Ξ~+1⋮⋮⋱⋮00⋯H~t-1,γt-1ƛΦt-1-1Γt-1,QΞ~≜diag⁡Q,Q,…,Q︷Ξ~,RΞ~≜diag⁡R~,R~,…,R~︷Ξ~,Ϝt-gt-h≜∏i=ghΦt-i-1=Φt-g-1Φt-g+1-1⋯Φt-h-1.The noise terms HΞ~,γ(s)Wt-1+Vt-1,γ(s) in (37) can be expressed as zero mean with covariance ΘΞ~,γ(s) by(39)Θi,γs=Hi,γsdiag⁡Q,Q,…,Q︷iHi,γsT+diag⁡R~,R~,…,R~︷i.Substituting (37) into (33) and taking the expectation on both sides of (33), the following relations can be obtained:(40)x^t∣t-1=H-Y~t-1ƛ=H-FΞ~,γsxt-Ξ~+H-HΞ~,γsWt-1+Vt-1,γs,Ex^t∣t-1=H-FΞ~,γsExt-Ξ~,with the horizon initial conditions(41)x^t-Ξ~=ϖt-Ξ~=Ext-Ξ~,P~t-Ξ~=P~t-Ξ~=covxt-Ξ~,xt-Ξ~determined by Lyapunov equation on the horizon k∈[t0,tt-Ξ~) as(42)ϖk+1=Φkϖk,ϖt0=ϖ0,P~k+1=ΦkP~k+ΦkT+Q~k,P~t0=P~0.Consider the unbiased condition; that is,E[x^t∣t-1]=E[xt-Ξ~], and, irrespective of the initial states, constraint (43) is required:(43)H-FΞ~,γs=I.Thus, the objective now turns to obtain the optimal gain matrixH-o with the unbiasedness constraint (43), such that the estimation error et of the estimate x^t∣t-1 has the following minimum variance:(44)H-o=arg⁡minH-⁡EetTet=arg⁡minH-⁡Etr⁡xt-x^t∣t-1Txt-x^t∣t-1=arg⁡minH-⁡HΞ~,γsWt-1+Vt-1,γsTH-TH-HΞ~,γsWt-1+Vt-1,γs.By using Lemma7, the gain matrix H~ can be obtained by the following correspondences:(45)A⟵HΞ~,B⟵O,C⟵QΞ~,D⟵RΞ~,E⟵FΞ~,γs,F⟵I.Thus, when{Φk,H~k,γk[ƛ]}(k∈[t-Ξ~,t-1]) is observable and Ξ[ƛ]+ħ≥n, the DFIR for the reorganized system (20) can be given as (46)x^tt-1ƛ=H-oY~t-1ƛ=I0FΞ~,γsTΘΞ~,γs-1FΞ~,γs-1FΞ~,γsTΘΞ~,γs-1∗Y~t-1ƛ=FΞ~,γsTΘΞ~,γs-1FΞ~,γs-1FΞ~,γsTΘΞ~,γs-1Y~t-1ƛ.Remark 8.
The batch form of RHE (44) requires the inversion computation of matrices ΘΞ~,γ(s)-1 and FΞ~,γ(s)TΘΞ~,γ(s)-1FΞ~,γ(s), whose dimension will become large as the horizon length Ξ[ƛ] increases; besides, the singularity problem will be encountered due to the impacts of LOS/NLOS condition.Remark 9.
The FIR filter uses finite measurements of the recent time interval. The length of the interval is called horizon size, which is also referred to as the window size or memory size. In (13), the matrices with subscript “Ξ[ƛ]” are obtained using the information on the recent time interval [t-Ξ[ƛ]-ħ,t-1], where the horizon size is Ξ[ƛ]+ħ. The large matrices defined in (13) are time-varying.


## 3.1. Delayed Observation Reorganization

In our research, the remote sensing observations (i.e.,{zt[i],xt[i]}) are impacted by delay and complementary issues and thus should not be adopted directly by the estimators. In this subsection, we focus on the delay issue; more specifically, we propose a preestimating scheme, including two steps, to reorganize the delay observations without dimensional augment.Step 1.
We rearrange the complementary observations{y[i](k)t-Ξ[i]t,i∈Nt[ƛ]}, which are received in horizon [t-Ξ[ƛ],t], that is, (8), and rewrite them into two parts with the batch form as(14)y~ħs≜y0jTsj=1l,yħjTs+ħj=1lT,t-Ξs-ħ≤s≤t-ħ,y~t-ss≜y0jTsj=1l,yt-sjTtj=1lT,t-ħ<s≤t.
Then, reorganized observationy~(·) is delay-free and satisfies(15)y~ħs≜H~ħsxs+v~ħs,γs,s+ħi,t-Ξs-ħ≤s≤t-ħ,y~t-ss≜H~t-ssxs+v~t-ss,γs,ti,t-ħ<s≤t,where(16)H~ħs=κs,0iϕsii=1lTκs+ħ,ħiϕs+hii=1lTT,H~t-ss=κs,0iϕsii=1lTκt,t-siϕtii=1lTT,v~ħs,γs,s+ħi=κs,0iusiγsii=1lTκs+ħ,ħius+ħiγs+ħii=1lTT,v~t-ss,γs,ti=κs,0iusiγsii=1lTκt,t-siutiγtii=1lTT,with vħ(s) and vt-s(s) being white noises of zero means and covariance matrices:(17)R~ħs=diag⁡κs,0iRii=1l,κs+ħ,ħiRii=1l,R~t-ss=diag⁡κs,0iRii=1l,κs+ħ,ħiRii=1l.Moreover, we can follow the result in [1] that the following lemma is true.Lemma 4.
For the given time horizon[t-Ξ[ƛ],t], the linear space L1≜{y(k)}k=t-Ξ[ƛ]t generated by {y(k)}k=t-Ξ[ƛ]t is equivalent to the linear space of(18)L2≜y~ħss=t-Ξƛ-ħt-ħ-1;y~t-sss=t-ħt, which implies that L2 contains the same information as L1.Step 2.
We further fuse the complementary observations (15) of the same stamp s∈[t-Ξ[ƛ]-ħ,t) into one complete measurement y~m[ƛ](s) (m∈{ħ,t-s}) prior to the local estimation process, and thus(19)y~mƛs=R-·∑i=1lgy~mis,xmis,m∈h,t-s,s=t-Ξƛ-ħ,…,t, where R- denotes the weighted matrix; g(y~m[i](s),xm[i](s)) is the relative measure function. Thus, the local measurements y~m[ƛ](s)∈Rny in (19) can be rewritten in the reorganized linear space L2 as(20)y~mƛs=ϑsƛH~mƛsxs+v~mƛs,γmƛ, where ϑs[ƛ] represents the failed fusion of y~[ƛ](s) at time stamp s=t-Ξ[ƛ]-ħ,…,t-1, which takes the value of 0 or 1, and is modeled as Bernoulli distributed white sequences satisfying Probϑs[ƛ]=1=ϑ-[ƛ]; 0<ϑ-[ƛ]≤1.Remark 5.
Note thatϑs[ƛ] (s=t-Ξ[ƛ]-ħ,…,t-1) is introduced to represent the failure of the preestimating algorithm (19) at time s, with the reason that the numbers of reorganized CMs y~ħ∣t(s) or y~t-s∣t(s) are less than a lower limit [9, 12]. Thus, through preestimating transformation, the remote sensing CMs in (14) with high dimension are transformed into the data missing in (20) with lower dimension.
Besides,R- and g(y~m[i](s),xm[i](s)) in (19) will be deduced through a complementary fusion strategy, with the details as follows.


## 3.2. Complementary Fusion Strategy

Let{y~m[i](s),i=1,2,…,l} be the set of complementary observations generated at instant s, the relative measures between objective ƛ and its neighbors i∈Ns[ƛ], that is, relative distance and relative angle, are calculated through RSS and AOA, respectively, as depicted in (4), and the measure noise covariance R~m[i](s) in (17).Hereafter, the fused measurement covarianceR- and the fused measurement variable ym[ƛ](s) (m∈{ħ,t-s}) can be expressed as(21)R-=∑i=1lR~mis-1-1,m∈ħ,t-s(22)ymƛs=R-·∑i=1lR~mis-1y~mis+xmls-xmis.Besides, the sensitivity matrix of the fused measurement variable (22) can be written as(23)H-=R-·∑i=1lR~mis-1Hi;hereinto, H[i] is the sensitivity matrix corresponding to y~m[i](s):(24)Hi=-I3×303×3.Note that the estimation error in this research may not always fall within the confidence level of the predicted covariance boundary. To avoid this problematic scenario, we introduce a new weighting factorψi,y as follows.Definition 6.
Given the CMsy~m[i](s), m∈{ħ,t-s}, sensitivity matrix H[i], and the inverse measurement noise variance, (R~m[i](s))-1, the weighting factor, ψi,y, is applied before they are fused; more specifically, it is defined as(25)ψi,y=ϱi2ϱTϱ,where ∑i=1lψi,y=1; ϱ=ϱ1ϱ2⋯ϱlT.The weighting factorψi,y is a function of (4). For instance, RSS vector is defined as Pm,rss=y~m,rss1⋯y~m,rsslT, m∈{ħ,t-s}. In the proposed weight strategy, the weighting factor of each CM is assigned based on the relative measures between i∈Ns[ƛ] and ƛ. Lower weight is assigned to the neighbors i∈Ns[ƛ] closer to ƛ. In other words, weighting factor ψi,y is selected to be proportional to the relative measures between the neighbors. The last measure y~m,rss[l], which is equivalent to max(Pm,rss), has highest ψi,y. Define the terms ΔPm,rss and ϱ as(26)ΔPm,rssi=y~m,rssl-y~m,rssi,m∈ħ,t-s,ϱi=1-ΔPm,rssiΔPm,rssi,m∈ħ,t-s,where ΔPm,rss=[ΔPm,rss[1],ΔPm,rss[2],…,ΔPm,rss[l]].Thus, the newly fused measurement noise covariance in (21) can be rewritten as(27)R-=l·∑i=1lψi,yR~mis-1-1,m∈ħ,t-s.Furthermore, the sensitivity matrix in (23) can be rewritten as(28)H-=∑i=1lψi,yR~mis-1-1∑i=1lψi,yR~mis-1Hi,m∈ħ,t-s.Meanwhile, the newly fused measurement vector in (22) can be rewritten as(29)ymƛs=R-·∑i=1lgy~mis,xmis=l·∑i=1lψi,yR~mis-1-1∑i=1l1l·ψi,yR~mis-1y~mis+xmls-xmis,m∈ħ,t-s,where(30)gy~mis,xmis=1l·ψi,yR~mis-1y~mis+xmls-xmis,m∈ħ,t-s.Note that the resulting problem is solved by a robust recursive scheme, which allows one to determine approximate estimates with a reduced computational condition. The details are presented in the following subsection.


## 3.3. Delay-Aware FIR Estimation Algorithm

In this subsection, we proposed a novel unbiased FIR estimator, namely, delay-aware FIR (DFIR) estimation algorithm, to realize the network-based state estimation of a generic nodeƛ∈Cg based on the cooperative observation of its neighborhood Ns[ƛ]. Note that several unique network-induced constraints are involved, including complementary observations, Markov jump parameters, and random delay; hereinto, partial constraints have been transformed in Sections 3.1 and 3.2, which may simplify the design procedure of DFIR.Prior to the main result of this section, let us recall a useful lemma that will be useful in DFIR algorithm.Lemma 7 (see [24]).
Suppose that the general trace optimization (GTO) problem is given as(31)minH-tr⁡H-A-BCH-A-BT+H-DH-T,subject  toH-E=F,where A, B, C, D, E, and F denote the constant matrices with appropriate dimensions: C=CT>0 and D=DT>0. tr(J) is the trace of the main diagonal of J.Thus, the unique solution to the GTO problem can be obtained as(32)H-=FBETΛ-1E-1ETΛ-1CATΛ-1I-EETΛ-1E-1ETΛ-1,where Λ=ACAT+D.On the recent finite-time horizon[t-Ξ[ƛ]-ħ,t-1] of reorganized linear space L2, the receding horizon filter for (3) and (20) can be written as (33)x^tt-1ƛ=∑k=t-Ξƛ-ħt-1H~t-k,γsy~ħƛs=H~γsY~t-1ƛ, where H~γ(s) is the gain matrix, Y~t-1[ƛ] is the augmented measurement vector, and γ(s) is the set of switching factors on the horizon [t-Ξ[ƛ]-ħ,t-1] defined by (34)–(36), respectively, as(34)H~γs≜H~Ξƛ+ħ,γsH~Ξƛ+ħ-1,γs⋯H~1,γs,(35)Y~t-1ƛ≜y~t-Ξƛ-ħƛTy~t-Ξƛ-ħ+1ƛT⋯y~t-1ƛTT,(36)γs≜γt-Ξƛ-ħƛ,γt-Ξƛ-ħ+1ƛ,…,γt-1ƛ.For ease of notation only, we express the horizon size of estimatorΞ[ƛ]+ħ as Ξ~; that is, Ξ~≜Ξ[ƛ]+ħ, Y~t-1[ƛ] can be rewritten in the batch form as(37)Y~t-1ƛ=FΞ~,γsxt-Ξ~+HΞ~,γsWt-1+Vt-1,γs, where(38)FΞ~,γs≜H~t-Ξ~,γt-Ξ~ƛϜt-Ξ~t-1⋮H~t-Ξ~+1,γt-Ξ~+1ƛϜt-2t-1H~t-1,γt-1ƛΦt-1-1,Wt-1≜wt-Ξ~wt-Ξ~+1⋮wt-1,Vt-1,γs≜v~t-Ξ~,γt-Ξ~ƛv~t-Ξ~+1,γt-Ξ~+1ƛ⋮v~t-1,γt-1ƛ,HΞ~,γs≜H~t-Ξ~,γt-Ξ~ƛΦt-Ξ~-1Γt-Ξ~H~t-Ξ~,γt-Ξ~ƛϜt-Ξ~t-Ξ~+1Γt-Ξ~⋯H~t-Ξ~,γt-Ξ~ƛϜt-Ξ~t-1Γt-Ξ~0H~t-Ξ~+1,γt-Ξ~+1ƛΦt-Ξ~+1-1Γt-Ξ~+1⋯H~t-Ξ~+1,γt-Ξ~+1ƛϜt-Ξ~+1t-1Γt-Ξ~+1⋮⋮⋱⋮00⋯H~t-1,γt-1ƛΦt-1-1Γt-1,QΞ~≜diag⁡Q,Q,…,Q︷Ξ~,RΞ~≜diag⁡R~,R~,…,R~︷Ξ~,Ϝt-gt-h≜∏i=ghΦt-i-1=Φt-g-1Φt-g+1-1⋯Φt-h-1.The noise terms HΞ~,γ(s)Wt-1+Vt-1,γ(s) in (37) can be expressed as zero mean with covariance ΘΞ~,γ(s) by(39)Θi,γs=Hi,γsdiag⁡Q,Q,…,Q︷iHi,γsT+diag⁡R~,R~,…,R~︷i.Substituting (37) into (33) and taking the expectation on both sides of (33), the following relations can be obtained:(40)x^t∣t-1=H-Y~t-1ƛ=H-FΞ~,γsxt-Ξ~+H-HΞ~,γsWt-1+Vt-1,γs,Ex^t∣t-1=H-FΞ~,γsExt-Ξ~,with the horizon initial conditions(41)x^t-Ξ~=ϖt-Ξ~=Ext-Ξ~,P~t-Ξ~=P~t-Ξ~=covxt-Ξ~,xt-Ξ~determined by Lyapunov equation on the horizon k∈[t0,tt-Ξ~) as(42)ϖk+1=Φkϖk,ϖt0=ϖ0,P~k+1=ΦkP~k+ΦkT+Q~k,P~t0=P~0.Consider the unbiased condition; that is,E[x^t∣t-1]=E[xt-Ξ~], and, irrespective of the initial states, constraint (43) is required:(43)H-FΞ~,γs=I.Thus, the objective now turns to obtain the optimal gain matrixH-o with the unbiasedness constraint (43), such that the estimation error et of the estimate x^t∣t-1 has the following minimum variance:(44)H-o=arg⁡minH-⁡EetTet=arg⁡minH-⁡Etr⁡xt-x^t∣t-1Txt-x^t∣t-1=arg⁡minH-⁡HΞ~,γsWt-1+Vt-1,γsTH-TH-HΞ~,γsWt-1+Vt-1,γs.By using Lemma7, the gain matrix H~ can be obtained by the following correspondences:(45)A⟵HΞ~,B⟵O,C⟵QΞ~,D⟵RΞ~,E⟵FΞ~,γs,F⟵I.Thus, when{Φk,H~k,γk[ƛ]}(k∈[t-Ξ~,t-1]) is observable and Ξ[ƛ]+ħ≥n, the DFIR for the reorganized system (20) can be given as (46)x^tt-1ƛ=H-oY~t-1ƛ=I0FΞ~,γsTΘΞ~,γs-1FΞ~,γs-1FΞ~,γsTΘΞ~,γs-1∗Y~t-1ƛ=FΞ~,γsTΘΞ~,γs-1FΞ~,γs-1FΞ~,γsTΘΞ~,γs-1Y~t-1ƛ.Remark 8.
The batch form of RHE (44) requires the inversion computation of matrices ΘΞ~,γ(s)-1 and FΞ~,γ(s)TΘΞ~,γ(s)-1FΞ~,γ(s), whose dimension will become large as the horizon length Ξ[ƛ] increases; besides, the singularity problem will be encountered due to the impacts of LOS/NLOS condition.Remark 9.
The FIR filter uses finite measurements of the recent time interval. The length of the interval is called horizon size, which is also referred to as the window size or memory size. In (13), the matrices with subscript “Ξ[ƛ]” are obtained using the information on the recent time interval [t-Ξ[ƛ]-ħ,t-1], where the horizon size is Ξ[ƛ]+ħ. The large matrices defined in (13) are time-varying.


## 4. Adaptive Finite-Horizon Group Estimator

In this section, we introduce an adaptive horizon group (AHG) approach to manage the horizon size in DFIR estimator. Different from the previous horizon group shift policy [26], AHG policy is designed to adjust the horizon size of DFIR algorithm according to the dynamic changed cooperative neighbors and the concerned network constraints. The AHG algorithm was combined with the proposed DFIR estimator and obtained AFGE estimator. The detailed algorithm of estimating with AHG is presented below.The first step of estimation with the DFIR is to construct a group of horizon sizes (i.e., horizon group). The horizon group at times is denoted by Gs and is defined as(47)Gs=Ξ~∣Ξ~=2Ξsc+ζ×τsi-ħ,τsi=0,1,2,…,ħ,where Ξ~ is the horizon size, Ξsc is the center horizon size at time s, ζ is the interval between horizon sizes, and ħ+1 is the total number of horizon sizes in a horizon group. Note that τs[i] impliedly reflected the possible transmission delay from the cooperative neighbor i∈Ns[ƛ], and ħ denotes the tolerable maximum transmission delay, which is defined in Assumption 3.The second step is to execute the DFIR estimator repeatedly with the horizon sizes inGs, which produces a group of state estimates, x^s1,x^s2,…,x^sħ+1. The reliable estimate is then selected by maximizing the likelihood function of ymƛsgYymƛs∣x=x^sΞ~, where x^s(Ξ~) is the state estimate obtained from (46) with the horizon size Ξ~. Note that the density function of v~m[ƛ](s,γm[ƛ]) is gV(v~m[ƛ](s,γm[ƛ])), where(48)gVv~mƛs,γmƛ=12πqdet⁡R-exp⁡-12v~mƛs,γmƛTR--1v~mƛs,γmƛ.The state estimates are then evaluated based on their likelihoods, which can be computed as follows:(49)ξx^siΞ~=pymƛs∣x^siΞ~=12πqdet⁡R-×exp-12ymƛs-ϑsƛH~mƛsx^siΞ~TR--1ymƛs-ϑsƛH~mƛsx^siΞ~, where ξ(x^si(Ξ~)) is the likelihood of the state estimate x^si(Ξ~) and H~m[ƛ](s) is the measurement matrix defined in (20):(50)Jx^sΞ~=12ymƛs-ϑsƛH~mƛsx^siΞ~TR--1ymƛs-ϑsƛH~mƛsx^siΞ~.Thus, a reliable estimate is selected by minimizingJ(x^s(Ξ~)). We use J(x^s(Ξ~)) as a measure of accuracy of the estimation result. The smaller the cost J(x^s(Ξ~)) is, the more reliable the estimated state is.The final step is to change the horizon group, which is called the AHG. If we determineΞsc, Gs is determined automatically using (47). The horizon size that produces a reliable estimate at time s is used as the center horizon size at time s+1. Let x^s(Ξsi) be the state estimate obtained from the FIR estimator using the horizon size Ξsi. The AHG can then be described by the following equation:(51)Ξs+1c=arg⁡maxΞsc⁡ξx^sΞsi.In this way, the horizon group is moved (or shifted) toward the horizon size that produces good estimation performance. In the AHG algorithm, the horizon size varies over time. Thus, we impose the lower boundΞ~min and the upper bound Ξ~max on the horizon size. We now summarize the estimation algorithm with DFIR, as depicted in Figure 2, whereas the detailed algorithm is presented as follows.Figure 2
Adaptive finite-horizon group estimation scheme with AHG.Algorithm 10 (AFGE estimator).
Select design parameters of the AHG algorithm, such asħ, ζ, Ξ~min, and Ξ~max. Set the initial value of the center horizon size as Ξsc=Ξ~min. For s=Ξ~min+1,Ξ~min+2,…, perform the following:(a)
Construct horizon groupGs that contains ħ horizon sizes using definition (9) as follows:(52)Gs=2Ξsc-ħ,2Ξsc+ζ-ħ,…,2Ξsc+ζ×ħ-1-ħ,2Ξsc+ζ×ħ-ħ=Ξs1,Ξs2,…,Ξsi,…,Ξsħ+1.(b)
Using the horizon sizesΞsi (i=1,2,…,ħ+1) contained in Gs, perform the following:fori=1toħ+1do
ifΞsi<s and Ξ~min<Ξsi<Ξ~max;then
(i)
Obtainx^s(Ξsi) by performing DFIR estimator (Algorithm 10) using Ξsi as follows:(53)x^ss-1ƛΞsi=FΞsi+ħ,γsTΘΞsi+ħ,γs-1FΞsi+ħ,γs-1FΞsi+ħ,γsTΘΞsi+ħ,γs-1Y~s-1ƛΞsi,whereFΞsi+ħ,γ(s), ΘΞsi+ħ,γ(s), and Y~s-1[ƛ](Ξsi) can be obtained using the horizon size Ξsi and definition (37).(ii)
Compute the likelihood ofx^s(Ξsi) using (49) as(54)ξx^siΞsi=12πqdetR-×exp-12ymƛs-ϑsƛH~mƛsx^siΞsiTR--1ymƛs-ϑsƛH~mƛsx^siΞsi.endifendfor(c)
The output (i.e., state estimate) of DFIR at time s, denoted by x^s, is determined as(55)x^s=arg⁡maxx^sΞsi⁡ξx^sΞsi.(d)
The center size at times+1, denoted by Ξs+1c, is determined as follows:(56)Ξs+1c=arg⁡maxΞsc⁡ξx^sΞsi.


## 5. An Application to Clustered UAV Systems

### 5.1. Simulation Configurations and Metrics

In this section, we illustrate the results proposed in Sections3 and 4 by numerical simulations. The position and velocity of a generic UAV evolve according to the state-space model as follows:(57)xpt+1xvt+1=I3δt·I303×3I3xptxvt+0.5·δt2·13×1δt·13×1wt,where xp(t) and xv(t) denote the position and velocity of a generic UAV at time t, respectively, δt represents the sampling period, and δt=0.1s; and w(t) is a zero-mean white noise with variance Qw, Qw=0.01I6; the initial values are x^0∣0=[120,30,0,0,0,0]T and P0∣0=diag⁡252I3,0.12I3.Consider a fleet ofNΣ=75 UAVs deployed schematically over a field of [-5,5]×[-5,5]×[0,1] km. The initial network topology is depicted in Figure 3(a). The objective UAV is set to track a trajectory as in Figure 3(b) and in flight for 1000 sec.Figure 3
Initial settings of network topology and ideal trajectory. (a) Initial topology of the networked vehicles. (b) Ideal trajectory of objective vehicle.
(a) (b)The objective UAV updates its state essentially based on cooperative packets of its neighbors, which include the time stamps and the navigational state of the sender x^t[i]=[x^p[i]T(t)x^v[i]T(t)]T. The relative position and relative angle are measured by RSS and AOA methods, respectively, with the observation models as in (4).For RSS measurements, the transmission power is taken asK=20 dBm, the path loss exponent is λ=2.1, and α=32.44. The standard deviation for measurement noise of AOA is 0.1 degrees. The variances of LOS and NLOS measurement noises are taken as Rt,1[i]=64 and Rt,2[i]=100, respectively. The bias of the NLOS measurement error is assumed to be μt[i]=-5. The occurrence probability of NLOS measurement error is taken as PNLOS=0.5 for each neighbor. The transition probabilities of Markov chain λt[i] are set as π11(2)=π22(2)=0.9 and π12(2)=π21(2)=0.1. To reduce the computational cost, not all the cooperative measurements are used; instead, only l=5 strongest received signals are adopted for state update of the objective UAV.As to the random delay, it is assumed thatτ(t)∈{0,1,2,3}. τ(t) is a prior known to the estimator and its paths can be described by κt,τ[i](i=1,2,…,l) with the restriction κt+τ1,τ1[i]×κt+τ2,τ2[i]=0 if τ1≠τ2. Hereinto, it is assumed that τ(t) is of the probabilities(58)Probτt=0=0.4,Probτt=1=0.3,Probτt=2=0.2,Probτt=3=0.1.A distributed fusion receding horizon estimator (DRHE) in [25] is adopted to compare with DFIR in Section 3 and the AFGE in Section 4. Let RMSE denote the root mean square error for the estimation of xt[i]; that is, ((1/M)∑j=1M(xt[j]-x^t∣t[j])2)0.5, where M=500 is the Monte Carlo simulation runs.


### 5.2. Simulation Results and Analysis

In this subsection, several key properties of the proposed AFGE approach, that is, AHG scheme and estimation errors, are tested on the networked UAVs with the models in Section5.1. The design parameters of the AFGE algorithm are taken as ħ=3, ζ=1, Ξmin=2, and Ξmax=40.First, we verify that AHG improves the estimation performance of the DFIR estimator by adapting the horizon size. To this end, we compare the AFGE with DFIR and DRHE. While the DFIR uses a constant horizon size, the AFGE uses the adaptive horizon size. Figure4 depicts the RMSE comparisons among DFIR, DRHE, and AFGE for diverse horizon sizes in the rectilinear orbit determination problem. In addition, this figure shows the RMSE of the AFGE estimator. The DFIR estimator produces the minimum RMSE of 0.216 when the horizon size is 12, whereas the minimum RMSE of DRHE is 0.320, with the horizon size of 13. The RMSE of the AFGE is 0.196, which is smaller than that of DFIR and DRHE. AFGE outperforms the DFIR using the best constant horizon size. Thus, it is verified that AHG improves the estimation performance of the DFIR.Figure 4
RMSEs comparison of DFIR, DRHE, and AFGE estimators.Next, we consider the UAV navigation scenario with the initial settings as in Section5.1. Besides, we set the horizon size of DFIR to Ξ[ƛ]=12, according to the analysis in Figure 4. The simulation results can be found in Figures 5 and 6.Figure 5
Localization results comparison and the related parameters. (a) Localization trajectories based on DRHE, DFIR, and AFGE. (b) Number of corresponding observable neighbors. (c) Random delayτ(t)∈{0,1,2,3} and the transition of NOS/NLOS errors γ(t)∈{0,1}.
(a) (b) (c)Figure 6
Performance comparison of estimation results between DRHE, DFIR, and AFGE. (a) Localization results. (b) Velocity results.
(a) (b)Hereinto, Figure5(a) reports the localization results based on RSS/AOA relative measurement and the CPs of its neighbors, while the related parameters are recorded in Figures 5(b) and 5(c). It is clear from Figure 5(b) that the number of received CPs in each step is greater than four, which means the observations are sufficient for the geometric localization in three-dimension space [9]. One path of τ(t) is given in Figure 5(c), based on which the optimal filter can be obtained directly by using the scheme in Sections 3 and 4. As expected, the localization result in Figure 5(a) demonstrates the validity of the proposed AFGE algorithm and navigation scheme.More specifically, Figures6(a) and 6(b) present the comparison of estimation results on location and velocity errors, respectively. The estimators of DFIR in Section 3 and DRHE [25] are chosen as the comparison with our proposed AFGE algorithm.As shown in the simulation results, the convergence rate of the DRHE is faster than that of DFIR due to the fact that DRHE adopts a distributed fusion structure in the design of estimator; however, the convergence error of the DFIR is smaller than that of DRHE, which is partly due to its reorganization of delayed observations and is also due to the design of DFIR considering the Markov jumping of LOS/NLOS errors. Furthermore, the performance of AFGE outperforms DRHE and DFIR; the reason can be owed to its reorganization scheme for delayed observation on one hand and considering the Markov jumping of LOS/NLOS errors on the other hand. Besides, the proposed AFGE also benefits from the AHG policy, which can manage the horizon size, thus acquiring the fastest convergence rate with the smallest estimation error.


## 5.1. Simulation Configurations and Metrics

In this section, we illustrate the results proposed in Sections3 and 4 by numerical simulations. The position and velocity of a generic UAV evolve according to the state-space model as follows:(57)xpt+1xvt+1=I3δt·I303×3I3xptxvt+0.5·δt2·13×1δt·13×1wt,where xp(t) and xv(t) denote the position and velocity of a generic UAV at time t, respectively, δt represents the sampling period, and δt=0.1s; and w(t) is a zero-mean white noise with variance Qw, Qw=0.01I6; the initial values are x^0∣0=[120,30,0,0,0,0]T and P0∣0=diag⁡252I3,0.12I3.Consider a fleet ofNΣ=75 UAVs deployed schematically over a field of [-5,5]×[-5,5]×[0,1] km. The initial network topology is depicted in Figure 3(a). The objective UAV is set to track a trajectory as in Figure 3(b) and in flight for 1000 sec.Figure 3
Initial settings of network topology and ideal trajectory. (a) Initial topology of the networked vehicles. (b) Ideal trajectory of objective vehicle.
(a) (b)The objective UAV updates its state essentially based on cooperative packets of its neighbors, which include the time stamps and the navigational state of the sender x^t[i]=[x^p[i]T(t)x^v[i]T(t)]T. The relative position and relative angle are measured by RSS and AOA methods, respectively, with the observation models as in (4).For RSS measurements, the transmission power is taken asK=20 dBm, the path loss exponent is λ=2.1, and α=32.44. The standard deviation for measurement noise of AOA is 0.1 degrees. The variances of LOS and NLOS measurement noises are taken as Rt,1[i]=64 and Rt,2[i]=100, respectively. The bias of the NLOS measurement error is assumed to be μt[i]=-5. The occurrence probability of NLOS measurement error is taken as PNLOS=0.5 for each neighbor. The transition probabilities of Markov chain λt[i] are set as π11(2)=π22(2)=0.9 and π12(2)=π21(2)=0.1. To reduce the computational cost, not all the cooperative measurements are used; instead, only l=5 strongest received signals are adopted for state update of the objective UAV.As to the random delay, it is assumed thatτ(t)∈{0,1,2,3}. τ(t) is a prior known to the estimator and its paths can be described by κt,τ[i](i=1,2,…,l) with the restriction κt+τ1,τ1[i]×κt+τ2,τ2[i]=0 if τ1≠τ2. Hereinto, it is assumed that τ(t) is of the probabilities(58)Probτt=0=0.4,Probτt=1=0.3,Probτt=2=0.2,Probτt=3=0.1.A distributed fusion receding horizon estimator (DRHE) in [25] is adopted to compare with DFIR in Section 3 and the AFGE in Section 4. Let RMSE denote the root mean square error for the estimation of xt[i]; that is, ((1/M)∑j=1M(xt[j]-x^t∣t[j])2)0.5, where M=500 is the Monte Carlo simulation runs.


## 5.2. Simulation Results and Analysis

In this subsection, several key properties of the proposed AFGE approach, that is, AHG scheme and estimation errors, are tested on the networked UAVs with the models in Section5.1. The design parameters of the AFGE algorithm are taken as ħ=3, ζ=1, Ξmin=2, and Ξmax=40.First, we verify that AHG improves the estimation performance of the DFIR estimator by adapting the horizon size. To this end, we compare the AFGE with DFIR and DRHE. While the DFIR uses a constant horizon size, the AFGE uses the adaptive horizon size. Figure4 depicts the RMSE comparisons among DFIR, DRHE, and AFGE for diverse horizon sizes in the rectilinear orbit determination problem. In addition, this figure shows the RMSE of the AFGE estimator. The DFIR estimator produces the minimum RMSE of 0.216 when the horizon size is 12, whereas the minimum RMSE of DRHE is 0.320, with the horizon size of 13. The RMSE of the AFGE is 0.196, which is smaller than that of DFIR and DRHE. AFGE outperforms the DFIR using the best constant horizon size. Thus, it is verified that AHG improves the estimation performance of the DFIR.Figure 4
RMSEs comparison of DFIR, DRHE, and AFGE estimators.Next, we consider the UAV navigation scenario with the initial settings as in Section5.1. Besides, we set the horizon size of DFIR to Ξ[ƛ]=12, according to the analysis in Figure 4. The simulation results can be found in Figures 5 and 6.Figure 5
Localization results comparison and the related parameters. (a) Localization trajectories based on DRHE, DFIR, and AFGE. (b) Number of corresponding observable neighbors. (c) Random delayτ(t)∈{0,1,2,3} and the transition of NOS/NLOS errors γ(t)∈{0,1}.
(a) (b) (c)Figure 6
Performance comparison of estimation results between DRHE, DFIR, and AFGE. (a) Localization results. (b) Velocity results.
(a) (b)Hereinto, Figure5(a) reports the localization results based on RSS/AOA relative measurement and the CPs of its neighbors, while the related parameters are recorded in Figures 5(b) and 5(c). It is clear from Figure 5(b) that the number of received CPs in each step is greater than four, which means the observations are sufficient for the geometric localization in three-dimension space [9]. One path of τ(t) is given in Figure 5(c), based on which the optimal filter can be obtained directly by using the scheme in Sections 3 and 4. As expected, the localization result in Figure 5(a) demonstrates the validity of the proposed AFGE algorithm and navigation scheme.More specifically, Figures6(a) and 6(b) present the comparison of estimation results on location and velocity errors, respectively. The estimators of DFIR in Section 3 and DRHE [25] are chosen as the comparison with our proposed AFGE algorithm.As shown in the simulation results, the convergence rate of the DRHE is faster than that of DFIR due to the fact that DRHE adopts a distributed fusion structure in the design of estimator; however, the convergence error of the DFIR is smaller than that of DRHE, which is partly due to its reorganization of delayed observations and is also due to the design of DFIR considering the Markov jumping of LOS/NLOS errors. Furthermore, the performance of AFGE outperforms DRHE and DFIR; the reason can be owed to its reorganization scheme for delayed observation on one hand and considering the Markov jumping of LOS/NLOS errors on the other hand. Besides, the proposed AFGE also benefits from the AHG policy, which can manage the horizon size, thus acquiring the fastest convergence rate with the smallest estimation error.


## 6. Conclusions and Discussions

In this paper, an adaptive horizon estimator has been addressed to solve the state estimation of NNS with the features of remote sensing complementary observations (RSOs) under mixed LOS/NLOS environments. An adaptive finite-horizon group estimator (AFGE) has been proposed, where the horizon size can be adjusted in real time according to stochastic parameters and random delays. First, a delay-aware FIR (DFIR) estimator has been derived with observation reorganization and complementary fusion strategies. Second, an adaptive horizon group (AHG) policy has been proposed to manage the horizon size. The AFGE algorithm is thus realized by combining AHG policy and DFIR estimator. A numerical simulation example has been used to demonstrate the potential of the proposed approach to robust estimation in terms of both performances and computational tractability.Future research will focus on other typical issues of NNS in terms of multirate fusion problem and cooperative attitude estimation problem.



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*Source: 6489165-2016-12-20.xml*