# UFIR Filtering for GPS-Based Tracking over WSNs with Delayed and Missing Data

Authors: Karen Uribe-Murcia; Yuriy S. Shmaliy; Jose A. Andrade-Lucio
Journal: Journal of Electrical and Computer Engineering
 (2018)

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## Abstract

In smart cities, vehicles tracking is organized to increase safety by localizing cars using the Global Positioning System (GPS). The GPS-based system provides accurate tracking but is also required to be reliable and robust. As a main estimator, we propose using the unbiased finite impulse response (UFIR) filter, which meets these needs as being more robust than the Kalman filter (KF). The UFIR filter is developed for vehicle tracking in discrete-time state-space over wireless sensor networks (WSNs) with time-stamped data discretely delayed onk-step-lags and missing data. The state-space model is represented in a way such that the UFIR filter, KF, and H∞ filter can be used universally. Applications are given for measurement data, which are cooperatively transferred from a vehicle to a central station through several nodes with k-step-lags. Better tracking performance of the UFIR filter is shown experimentally.


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## Body

## 1. Introduction

Accurate target tracking is one of the key problems in urban areas [1], which especially arises in smart cities design [2]. If a target is equipped with the Global Positioning System (GPS) tracker, then measurement data can be transferred to a central station through one or several nodes of a wireless sensor network (WSN) [3]. The problem which arises here is associated with information latency and missing data [4] due to the following main causes: high maneuverability of the target [1], failures in measurements [5], network congestion [3], non-line-of-sight (NLOS) problems [6, 7], and accidental loss of some collected data [8]. Furthermore, latency naturally occurs due to the limited bandwidth, finite propagation time [9], complexity of very large-scale integration and microelectromechanical systems [10], and time required to complete operations such as signal conditioning and storage [11]. In networks, communication delays go along with data loss called dropout or intermittence [12, 13]. Also the delay between the measurement and its availability to the filter causes the problem of out-of-sequence measurement [6, 14].Two basic models have been created for delayed data. The delays are assumed to be known when sensors are able to detect the delays or data are time-stamping [15, 16]. In many other applications [17, 18], the delays are considered to be random. The problem becomes more complex in uncertain systems [19]. The best estimate is obtained here by combining delayed and nondelayed data with different probabilities.The Kalman andH∞ state estimators are most widely used to deal with latency and associated issues [20]. The linear Kalman filter (KF) is optimal when it matches the system perfectly, noise is white Gaussian and uncorrelated, and the noise statistics are known along with the initial values. When such conditions are not obeyed, the KF may demonstrate poor performance [13, 21]. The robust H∞ filter bounds the mean square error (MSE) for admissible parameter perturbations and delays [19, 22], which allows for minimizing errors with less information required than for the noise statistics [20, 23].Another way to achieve better robustness is to process most recent finite data [24] using finite impulse response (FIR) filters [25]. Such filters have been developed during decades by many authors in signal processing [26–33] and control [34–36]. However, only a few authors have proposed FIR solutions for models with delays [37–40]. Let us notice that the available iterative unbiased FIR (UFIR) algorithm [28, 41–43] is most robust among other FIR solutions owing to an ability to ignore the noise statistics and initial values. This filter is bounded-input bounded-output (BIBO) stable and blind on given horizons of N points, but is still not developed for observations with delayed and missing data.In this paper, we develop the UFIR filter for GPS-based vehicle tracking over WSNs with time-stamped data discretely delayed and missing data. The rest of the paper is organized as follows. In Section2, we consider the model and formulate the problem. In Section 3, we develop the UFIR filter for observations with delayed and missing data. Section 4 discusses the estimation errors. Section 5 gives an experimental example of applications to GPS-based tracking and concluding remarks are drawn in Section 6.


## 2. Tracking Model and Problem Formulation

A typical scenario of GPS-based vehicle tracking in WSNs is sketched in Figure1. The vehicle current coordinates are measured by the GPS tracker. The time-stamped data are transferred to a central station (CS) via one or several nodes of the WSN. Because each node may discretely delay time-stamped data at least on one-step, the vehicle location is observed in CS with a time varying kn-step-lag depending on the vehicle location and interaction with the WSN.Figure 1
Transferring the time-stamped vehicle coordinates measured by a GPS tracker to a central station (CS) via several nodes of a WSN. Latency with akn-step-lag is caused by delays in the node.


### 2.1. Tracking Model

Forkn⩾0, the vehicle dynamics and its observation can be represented in discrete-time state-space as(1)xn=Fxn-1+wn,(2)yn=Hxn-kn+vn,where n is the discrete-time index, xn∈RK is the vehicle state vector, yn∈RM is the observation vector, F∈RK×K is not singular, and H∈RM×K. All data are time-stamped, so that kn is known at each n. Regardless of the delay, the initial state xn-1 is supposed to be known. The uncorrelated noise vectors, wn∈RK and vn∈RM, are white Gaussian with known covariances, Q=E{wnwnT} and R=E{vnvnT}, and the property E{wnvqT}=0 for all n and q.The UFIR filter can be applied if to transform model (1)-(2) to have no latency. That can be done if to represent xn-kn using (1) via xn as(3)xn-kn=F-knxn-∑i=0kn-1Fiwn-iand then substitute (3) into (2) and arrive at(4)yn=H¯nxn+v¯n,where(5)H¯n=HF-kn,(6)v¯n=vn-H∑i=0kn-1F-kn+iwn-iand the covariance R¯n=E{v¯nv¯nT} of v¯n is given by(7)R¯n=R+H¯n∑i=0kn-1FiQFiTH¯nT.In compact matrix forms, (6) and (7) can be represented as(8)v¯n=vn-HB¯nWpn,n,(9)R¯n=R+HB¯nQ¯nB¯nTHT,where(10)B¯n=F-1F-2⋯F-kn,(11)Wpn,n=wpnTwpn+1T⋯wnTT,and Q¯n=diagQQ⋯Q has kn diagonal components. Here, B¯n=0 and Wpn,n=0 when kn=0 and pn>n.Any standard estimation technique can now be applied to models (1) and (4). However, the KF and H∞ filter were most developed for data with latency. Therefore, below we will introduce in brief only these filters and then compare them to the UFIR filter based on examples of tracking. We will use the following measures: x^n≜x^n∣n is the estimate of xn over data taken from past up to and including time index n, x^n- is the prior estimate, Pn=E{(xn-x^n)xn-x^nT} is the error covariance matrix, and Pn-=E{(xn-x^n-)xn-x^n-T} is the prior error covariance matrix.


#### 2.1.1. Kalman Filter

For our purposes, we will exploit an alternative form of the KF algorithm given in [44]. This algorithm starts with the prior error covariance matrix(12)Pn-=FPn-1FT+Qand then recursively updates the following values:(13)Pn=Pn--1+H¯nTR¯n-1H¯n,KnKF=Pn-1H¯nTR¯n-1,x^n=Fx^n-1+KnKFyn-H¯nFx^n-1,Pn+1-=FPn-1FT+Q,where H¯n is given by (5) and R¯n by (7) for any kn⩾0.


#### 2.1.2.H∞ Filter

TheH∞ filter has been derived in [44] in the form (13) of the KF using the game theory. For kn⩾0, the H∞ filtering algorithm becomes(14)Pn=Pˇn--1-θnSn+H¯nTRˇn-1H¯n,(15)Kn∞=Pn-1H¯nTRˇn-1,(16)x^n=Fx^n-1+Kn∞yn-H¯nFx^n-1,(17)Pˇn+1-=FPn-1FT+Qˇ,where the user-given symmetric positive definite matrices P0, Qˇ, and Rˇ have different meanings than in the KF and Pˇn- can be computed via P0 using (12) with Q=Qˇ. To keep (14) positive definite, the positive definite matrix Sn∈RK×K is subject to(18)Pˇn--1-θnSn+H¯nTRˇ-1H¯n>0.If equal weights are required for all errors, matrix Sn must be set identity, Sn=I. The performance criterion for this filter is Jn<1/θn, in which a scalar θn must be small enough in order for the filter to be efficient. It then follows that the tuning factor θn is not allowed to be negative, even though its negative values may reduce errors when the weighting matrices are not maximized. For Gaussian noise, zero θn transforms the H∞ filter to the KF. For any other noise, small θn>0 may result in better robustness.The problem now formulates as follows. Given (1) and (4) with time varying kn⩾0 and missing data, we would like to develop the UFIR filter and find its fast iterative form for GPS-based tracking of a moving vehicle as shown in Figure 1. We also wish to know how the UFIR filter, KF, and H∞ filter measure to each other in applications to tracking.


## 2.1. Tracking Model

Forkn⩾0, the vehicle dynamics and its observation can be represented in discrete-time state-space as(1)xn=Fxn-1+wn,(2)yn=Hxn-kn+vn,where n is the discrete-time index, xn∈RK is the vehicle state vector, yn∈RM is the observation vector, F∈RK×K is not singular, and H∈RM×K. All data are time-stamped, so that kn is known at each n. Regardless of the delay, the initial state xn-1 is supposed to be known. The uncorrelated noise vectors, wn∈RK and vn∈RM, are white Gaussian with known covariances, Q=E{wnwnT} and R=E{vnvnT}, and the property E{wnvqT}=0 for all n and q.The UFIR filter can be applied if to transform model (1)-(2) to have no latency. That can be done if to represent xn-kn using (1) via xn as(3)xn-kn=F-knxn-∑i=0kn-1Fiwn-iand then substitute (3) into (2) and arrive at(4)yn=H¯nxn+v¯n,where(5)H¯n=HF-kn,(6)v¯n=vn-H∑i=0kn-1F-kn+iwn-iand the covariance R¯n=E{v¯nv¯nT} of v¯n is given by(7)R¯n=R+H¯n∑i=0kn-1FiQFiTH¯nT.In compact matrix forms, (6) and (7) can be represented as(8)v¯n=vn-HB¯nWpn,n,(9)R¯n=R+HB¯nQ¯nB¯nTHT,where(10)B¯n=F-1F-2⋯F-kn,(11)Wpn,n=wpnTwpn+1T⋯wnTT,and Q¯n=diagQQ⋯Q has kn diagonal components. Here, B¯n=0 and Wpn,n=0 when kn=0 and pn>n.Any standard estimation technique can now be applied to models (1) and (4). However, the KF and H∞ filter were most developed for data with latency. Therefore, below we will introduce in brief only these filters and then compare them to the UFIR filter based on examples of tracking. We will use the following measures: x^n≜x^n∣n is the estimate of xn over data taken from past up to and including time index n, x^n- is the prior estimate, Pn=E{(xn-x^n)xn-x^nT} is the error covariance matrix, and Pn-=E{(xn-x^n-)xn-x^n-T} is the prior error covariance matrix.


### 2.1.1. Kalman Filter

For our purposes, we will exploit an alternative form of the KF algorithm given in [44]. This algorithm starts with the prior error covariance matrix(12)Pn-=FPn-1FT+Qand then recursively updates the following values:(13)Pn=Pn--1+H¯nTR¯n-1H¯n,KnKF=Pn-1H¯nTR¯n-1,x^n=Fx^n-1+KnKFyn-H¯nFx^n-1,Pn+1-=FPn-1FT+Q,where H¯n is given by (5) and R¯n by (7) for any kn⩾0.


### 2.1.2.H∞ Filter

TheH∞ filter has been derived in [44] in the form (13) of the KF using the game theory. For kn⩾0, the H∞ filtering algorithm becomes(14)Pn=Pˇn--1-θnSn+H¯nTRˇn-1H¯n,(15)Kn∞=Pn-1H¯nTRˇn-1,(16)x^n=Fx^n-1+Kn∞yn-H¯nFx^n-1,(17)Pˇn+1-=FPn-1FT+Qˇ,where the user-given symmetric positive definite matrices P0, Qˇ, and Rˇ have different meanings than in the KF and Pˇn- can be computed via P0 using (12) with Q=Qˇ. To keep (14) positive definite, the positive definite matrix Sn∈RK×K is subject to(18)Pˇn--1-θnSn+H¯nTRˇ-1H¯n>0.If equal weights are required for all errors, matrix Sn must be set identity, Sn=I. The performance criterion for this filter is Jn<1/θn, in which a scalar θn must be small enough in order for the filter to be efficient. It then follows that the tuning factor θn is not allowed to be negative, even though its negative values may reduce errors when the weighting matrices are not maximized. For Gaussian noise, zero θn transforms the H∞ filter to the KF. For any other noise, small θn>0 may result in better robustness.The problem now formulates as follows. Given (1) and (4) with time varying kn⩾0 and missing data, we would like to develop the UFIR filter and find its fast iterative form for GPS-based tracking of a moving vehicle as shown in Figure 1. We also wish to know how the UFIR filter, KF, and H∞ filter measure to each other in applications to tracking.


## 2.1.1. Kalman Filter

For our purposes, we will exploit an alternative form of the KF algorithm given in [44]. This algorithm starts with the prior error covariance matrix(12)Pn-=FPn-1FT+Qand then recursively updates the following values:(13)Pn=Pn--1+H¯nTR¯n-1H¯n,KnKF=Pn-1H¯nTR¯n-1,x^n=Fx^n-1+KnKFyn-H¯nFx^n-1,Pn+1-=FPn-1FT+Q,where H¯n is given by (5) and R¯n by (7) for any kn⩾0.


## 2.1.2.H∞ Filter

TheH∞ filter has been derived in [44] in the form (13) of the KF using the game theory. For kn⩾0, the H∞ filtering algorithm becomes(14)Pn=Pˇn--1-θnSn+H¯nTRˇn-1H¯n,(15)Kn∞=Pn-1H¯nTRˇn-1,(16)x^n=Fx^n-1+Kn∞yn-H¯nFx^n-1,(17)Pˇn+1-=FPn-1FT+Qˇ,where the user-given symmetric positive definite matrices P0, Qˇ, and Rˇ have different meanings than in the KF and Pˇn- can be computed via P0 using (12) with Q=Qˇ. To keep (14) positive definite, the positive definite matrix Sn∈RK×K is subject to(18)Pˇn--1-θnSn+H¯nTRˇ-1H¯n>0.If equal weights are required for all errors, matrix Sn must be set identity, Sn=I. The performance criterion for this filter is Jn<1/θn, in which a scalar θn must be small enough in order for the filter to be efficient. It then follows that the tuning factor θn is not allowed to be negative, even though its negative values may reduce errors when the weighting matrices are not maximized. For Gaussian noise, zero θn transforms the H∞ filter to the KF. For any other noise, small θn>0 may result in better robustness.The problem now formulates as follows. Given (1) and (4) with time varying kn⩾0 and missing data, we would like to develop the UFIR filter and find its fast iterative form for GPS-based tracking of a moving vehicle as shown in Figure 1. We also wish to know how the UFIR filter, KF, and H∞ filter measure to each other in applications to tracking.


## 3. UFIR Filter for Tracking with Delayed and Missing Data

To develop the UFIR filter forkn>0, we extend models (1) and (4) on a horizon [m,n] of N points, from m=n-N+1 to n that referring to [43] yields(19)Xm,n=ANxm+BNWm,n,(20)Ym,n=Cm,nxm+Gm,nWm,n+Vm,nwith the following extended vectors and matrices:(21)Xm,n=xmTxm+1T⋯xnTT,(22)Ym,n=ymTym+1T⋯ynTT,(23)Vm,n=vm-CB¯nWpm,mvm+1-CB¯nWpm+1,m+1⋮vn-1-CB¯nWpn-1,n-1vn-CB¯nWpn,n,(24)AN=IFT⋯FN-1TT,(25)BN=I0⋯00FI⋯00⋮⋮⋱⋮⋮FN-2FN-3⋯I0FN-1FN-2⋯FI,(26)Cm,n≜Cm,nk=C¯m,nkAN,(27)Gm,n≜Gm,nk=C¯m,nkBN,(28)C¯m,nk=diagCF-kmCF-km+1⋯CF-kn︸N,where k represents a set of {km,km+1,…,kn}.


### 3.1. Batch UFIR Filter Form

The UFIR filtering estimatex^n of the vehicle state xn can be obtained at n in the batch form as [27, 34](29)x^n=Hm,nYm,n,where Hm,n is the UFIR filter gain and Ym,n is a vector of real data (22), if to satisfy the unbiasedness condition(30)Exn=Ex^n,in which E{z} means averaging of z and xn can be represented with the last row vector in (19) as(31)xn=FN-1xm+BNNWm,n,where BN(N) is the Nth row vector in (25) given by(32)BNN=FN-1FN-2⋯FI.By combining (29)–(32) and following [27], one arrives at the unbiasedness constraint(33)I=Hm,nCm,n,in which(34)Cm,n≜Cm,nk=CF-N+1-km⋮CF-1-kn-1CF-kn.Now multiplying the both sides of (33) with (Cm,nTCm,n)-1Cm,nTCm,n yields the UFIR filter gain(35)Hm,n=Cm,nTCm,n-1Cm,nTand the batch UFIR filtering estimate becomes(36a)x^n=Cm,nTCm,n-1Cm,nTYm,n(36b)=GnCm,nTYm,n,where thegeneralized noise power gain (GNPG) [43] is(37)Gn=Hm,nHm,nT=Cm,nTCm,n-1.The batch form (36a) may not suite real-time tracking and we go on with its fast iterative algorithm.


### 3.2. Iterative UFIR Filter Form

Provided (1) and (4), the standard iterative UFIR filtering algorithm [25] can be applied straightforwardly, if to substitute matrix H with H¯ given by (5). The UFIR filtering estimate (36a) can then be computed iteratively using recursions(38)Gl=H¯lTH¯l+FGl-1FT-1-1,(39)x^l=Fx^l-1+GlH¯lTyl-H¯lFx^l-1,beginning with l=m+K and ending when l=n. The initial values for (38) and (39) are obtained at s=m+K-1 in the batch forms as(40)Gs=Cm,sTCm,s-1,(41)x~s=GsCm,sTYm,s,where Ym,s is a vector (22) of real data. When some data are lost, inaccurate, or unavailable, yn can be predicted as yn=HFx^n-1, in which case first data on the horizon [0,N-1] must be available.It is known that the linear UFIR filter is BIBO stable and not prone to divergence. However, latency in information delivery may require an ability to predict lost values that inevitably cause extra tracking errors, which we will consider next.


## 3.1. Batch UFIR Filter Form

The UFIR filtering estimatex^n of the vehicle state xn can be obtained at n in the batch form as [27, 34](29)x^n=Hm,nYm,n,where Hm,n is the UFIR filter gain and Ym,n is a vector of real data (22), if to satisfy the unbiasedness condition(30)Exn=Ex^n,in which E{z} means averaging of z and xn can be represented with the last row vector in (19) as(31)xn=FN-1xm+BNNWm,n,where BN(N) is the Nth row vector in (25) given by(32)BNN=FN-1FN-2⋯FI.By combining (29)–(32) and following [27], one arrives at the unbiasedness constraint(33)I=Hm,nCm,n,in which(34)Cm,n≜Cm,nk=CF-N+1-km⋮CF-1-kn-1CF-kn.Now multiplying the both sides of (33) with (Cm,nTCm,n)-1Cm,nTCm,n yields the UFIR filter gain(35)Hm,n=Cm,nTCm,n-1Cm,nTand the batch UFIR filtering estimate becomes(36a)x^n=Cm,nTCm,n-1Cm,nTYm,n(36b)=GnCm,nTYm,n,where thegeneralized noise power gain (GNPG) [43] is(37)Gn=Hm,nHm,nT=Cm,nTCm,n-1.The batch form (36a) may not suite real-time tracking and we go on with its fast iterative algorithm.


## 3.2. Iterative UFIR Filter Form

Provided (1) and (4), the standard iterative UFIR filtering algorithm [25] can be applied straightforwardly, if to substitute matrix H with H¯ given by (5). The UFIR filtering estimate (36a) can then be computed iteratively using recursions(38)Gl=H¯lTH¯l+FGl-1FT-1-1,(39)x^l=Fx^l-1+GlH¯lTyl-H¯lFx^l-1,beginning with l=m+K and ending when l=n. The initial values for (38) and (39) are obtained at s=m+K-1 in the batch forms as(40)Gs=Cm,sTCm,s-1,(41)x~s=GsCm,sTYm,s,where Ym,s is a vector (22) of real data. When some data are lost, inaccurate, or unavailable, yn can be predicted as yn=HFx^n-1, in which case first data on the horizon [0,N-1] must be available.It is known that the linear UFIR filter is BIBO stable and not prone to divergence. However, latency in information delivery may require an ability to predict lost values that inevitably cause extra tracking errors, which we will consider next.


## 4. Tracking Errors Caused by Latency,kn⩾0

Error produced by the UFIR tracker can be defined asϵn=xn-x^n, where xn is specified by (31). Provided ϵn, the error covariance matrix Pn=E{ϵnϵnT} can also be represented in two forms.In the batch form, matrixPn appears if to substitute xn with (31) and employ x^n=Hm,nYm,n with Ym,n given by (20). That yields(42)Pn=BNN-Hm,nGm,nQ¯NBm,nN-Hm,nGm,nT+Hm,nR¯NHm,nT,where Q¯N=diagQQ⋯Q and R¯N=diagRR⋯R are square matrices with N nonzero diagonal elements. It can be shown that the deterministic case of Q¯N=0 and R¯N=0 makes Pn=0 and the UFIR tracker has thus thedeadbeat property.


### 4.1. Iterative Computation ofPn

Matrix (42) can also be computed iteratively, if to substitute xn with (1) and x^n with (39). Provided the averaging, the recursion for (42) can be found as(43)Pl=I-GlH¯lTH¯lPl-⋯T+GlH¯lTH¯l×∑i=0kl-1FiQFiTH¯lTH¯lGl+GlH¯lTRH¯lGl,where Pl- is given by (12), l ranges as in (38) and (39), and Pn is taken when l=n. Recursion (43) suggests that the tracking error grows with kn, because the sum containing Q grows with kn. However, the same cannot be said about R, which does not accumulate the effect of kn.With no latency, the sum in (43) becomes identically zero and one arrives at the error covariance Pl of the standard UFIR filter [43],(44)Pl=I-GlHTHPl-⋯T+GlHTRHGl,which also holds for the KF, if to substitute GnHT with the Kalman gain Kn.Note that the minimization of tracking errors will require an optimal numberNopt-1 of iterations for the UFIR filter. At the test stage, the optimal horizon Nopt can be found for the known ground truth xn by minimizing the MSE via the trace of Pn [30] depicted as tr Pn. Because the ground truth is unavailable in real tracking, Nopt can be estimated via the measurement residual as shown in [30].


## 4.1. Iterative Computation ofPn

Matrix (42) can also be computed iteratively, if to substitute xn with (1) and x^n with (39). Provided the averaging, the recursion for (42) can be found as(43)Pl=I-GlH¯lTH¯lPl-⋯T+GlH¯lTH¯l×∑i=0kl-1FiQFiTH¯lTH¯lGl+GlH¯lTRH¯lGl,where Pl- is given by (12), l ranges as in (38) and (39), and Pn is taken when l=n. Recursion (43) suggests that the tracking error grows with kn, because the sum containing Q grows with kn. However, the same cannot be said about R, which does not accumulate the effect of kn.With no latency, the sum in (43) becomes identically zero and one arrives at the error covariance Pl of the standard UFIR filter [43],(44)Pl=I-GlHTHPl-⋯T+GlHTRHGl,which also holds for the KF, if to substitute GnHT with the Kalman gain Kn.Note that the minimization of tracking errors will require an optimal numberNopt-1 of iterations for the UFIR filter. At the test stage, the optimal horizon Nopt can be found for the known ground truth xn by minimizing the MSE via the trace of Pn [30] depicted as tr Pn. Because the ground truth is unavailable in real tracking, Nopt can be estimated via the measurement residual as shown in [30].


## 5. GPS-Based Tracking of a Moving Vehicle

We will now consider the case shown in Figure1 when the GPS tracker measures the vehicle coordinates of location at each time index n and transfer time-stamped data cooperatively to a CS of a WSN via one or several nodes. We admit that each node may introduce latency and data will thus arrive at the CS with known delay on kn>0 points. At different time instances, a vehicle may interact with a different number of the nodes that will make the kn-step-lag time varying. We will base our investigations on data obtained in the Cook county of Illinois and available for free use from the University of Illinois at Chicago. To simplify the problem, in this paper we will consider the case of a constant latency, k=kn.Concerned with the tracking errors and not with the actual vehicle location, we will conventionally place the start point at zero coordinates as shown in Figure2.Figure 2
GPS-based vehicle trajectory measured in the north (y) and east (x) coordinates, both in km, with the start point at {0,0}. Measurements are provided each second at 858 data points.


### 5.1. State-Space Model

To investigate the trade-off between the estimators, we will suppose that a vehicle is represented with two states in each directions and assign the state vector,K=4, as xn=x1nx2nx3nx4nT, where x1n=xn, x2n=x˙n, x3n=yn, and x4n=y˙n. Accordingly, the system matrix attains the form of(45)F=1τ000100001τ0001, where the sampling time is τ=1 s for the considered database. The GPS tracker provides measurements of the vehicle coordinates, x and y. Therefore, the measurement matrix is(46)H=10000010.ProvidedF and H, matrix Cm,s required by the UFIR filter to compute the initial values (40) and (41) for s=m+3 becomes(47)Cs,m=1-3τ00001-3τ1-2τ00001-2τ1-τ00001-τ10000010F-k.The only tuning factorNopt=5 required by the UFIR filter was found for k=0 by minimizing the derivative of the trace of the mean square value of the residual yn-Hx^n(N), as shown in [30]. Because Nopt depends on k, we will apply Nopt=5 in the worst case for the UFIR filter.Having no information about the process noise, we observe similar trajectories and estimate the average vehicle speed by about 10 m/s or 36 km/hour. Next, accepting the speed standard deviation of about 20%, we setσw2=2 m/s to the second state along each of the coordinates, ignore the unknown noise in the first state, σw1=0, and describe matrix Q as(48)Q=σw22τ2/2τ/200τ/210000τ2/2τ/200τ/21.The GPS standard positioning service provides navigation with an error of less than 15 meters with the probability of 95% in the 2-sigma sense. Referring to this value, we assign the standard deviation of the measurement noise in each direction asσv=15/4=3.75 m and obtain(49)R=σv200σv2.Because the above provided matricesQ and R are overestimated, we set Qˇ=Q and Rˇ=R for H∞. It is expected that the tuning factor θ will improve the performance of the H∞ filter by minimizing the MSE for the maximized errors. However, the ground truth is not available in tracking. Therefore, we will find θ for the measured trajectory and consider it as the best case for H∞, which is unfeasible.


### 5.2. Effect of Latency on the Estimation Accuracy

We start with learning the effect ofk on the estimation accuracy, which is illustrated in Figure 3 with the root MSEs (RMSEs) in the north direction (y-RMSE). The KF is self-tuned to k. Therefore, we consider its RMSE as a benchmark. A special feature of the KF is that the RMSE grows with k nonlinearly and faster than in the UFIR and H∞ filters.Figure 3
Effect of thek-step-lag on the y-RMSE of the UFIR filter, KF, and H∞ filter with different tunings.The UFIR filter produces a bit more errors than in the KF with smallk and lesser with larger k. A special feature is that the UFIR estimate is of low sensitivity to N, in which optimal value Nopt=5 holds for 0⩽k⩽4, increases to Nopt=6 for 5⩽k⩽6, and reaches Nopt=7 when 7⩽k⩽10. Of practical importance is that setting N optimally for each k does not improve the performance essentially against the worst case when Nopt=5 is set for all k.TheH∞ filter outperforms both the UFIR filter and KF, provided that θ is set properly for each lag k. However, this filter is highly sensitive to θ, in which optimal value θopt ranges from 1.8×10-2 for k=0 to 1.066×10-5 for k=10 in a nonlinear way. Unlike in the UFIR filter, a constant θ is unacceptable for all k. An example is given in Figure 3, where θopt=7.1×10-4 found for k=2 is applied in a wide range of k. As can be seen, it is only when k=2 that the H∞ filter improves the KF performance. For k<2, there is no improvement and, when k>2, the H∞ filter rapidly diverges.


### 5.3. Tracking over Data Delayed onk=3

We now suppose that data are transferred from a vehicle to a CS withk=3 and investigate tracking errors in the north and east directions. Filters will be tuned as follows: UFIR in the worst case of Nopt=5 being valid for k=0; KF as near optimal; and H∞ being in the best (unfeasible) case of known ground truth.


#### 5.3.1. Tracking in the North Direction

The vehicle trajectory (measured and delayed) and estimates provided by the filters in the north direction are sketched in Figure4. As can be seen in Figure 4(a), the trajectory is nonlinear and the vehicle sometimes maneuvers rapidly. All filters produce consistent estimates with poorly distinguishable differences (Figure 4(a)). We therefore additionally show two parts of the trajectory on short-time spans of 0⩽n⩽200 in Figure 4(b) and 300⩽n⩽400 in Figure 4(c). Several observations can be made from Figure 4:(i)
All filters track well the trajectory when a vehicle travels with a near constant velocity in one direction, as on140⩽n⩽180 in Figure 4(b).(ii)
Filters temporarily lose an ability of tracking and go along the delayed data when a vehicle quickly changes the direction, as on340⩽n⩽343 in Figure 4(c).(iii)
Responding to fast maneuvers, all filters produce dynamic errors such that the UFIR filter comes up with larger excursions but shorter transients, KF with shorter excursions but longer transients, andH∞ filter demonstrates inbetween properties; see on 80⩽n⩽100 in Figure 4(b) and 344⩽n⩽360 in Figure 4(c).Figure 4
Tracking in the north directiony, m for k=3 with UFIR filter (Nopt=5), KF, and H∞ filter (θ=3.3×10-4): (a) full time scale, (b) 0⩽n⩽200, and (c) 300⩽n⩽400.
(a) (b) (c)Because errors are unacceptably large in all filters when they temporarily lose an ability of tracking, a key question arises of how fast each of them returns back to the normal mode. In this regard the UFIR filter looks better with its shortest transient.


#### 5.3.2. Tracking in the East Direction

Tracking in the east direction (Figure5) does not reveal any essential features. The filters still track well the trajectory when a vehicle travels with a near constant velocity as, for example, in a span of 120⩽n⩽160 in Figure 5(b). Here, all filters also temporarily lose an ability of tracking and go along the delayed data when the trajectory quickly changes as, for example, in a span of 560⩽n⩽570 in Figure 5(c). Finally, the UFIR filter still demonstrates larger excursions but shorter transients, KF shorter excursions but longer transients, and H∞ filter is inbetween as, for example, in spans of 60⩽n⩽80 in Figure 5(b) and 570⩽n⩽580 in Figure 5(c).Figure 5
Tracking in the east directionx, m for k=3 with UFIR filter (Nopt=5), KF, and H∞ filter (θ=3.3×10-4): (a) full time scale, (b) 0⩽n⩽200, and (c) 500⩽n⩽600.
(a) (b) (c)


### 5.4. Tracking with Temporary Lost Data

We finally admit that some data points can be lost during the transmission and remove 5 data points atn=(615⋯620) s and 10 at n=(665⋯675) s as shown in Figure 6 for k=3. To predict lost data, we augment each algorithm with the prediction block as mentioned below (41) and run the filters. As can be seen, the filters act consistently with, however, some specifics. The estimates do not get away essentially from each other and the actual trajectory. However, when a vehicle maneuvers during the prediction, all filters diverge and return back to the actual trajectory with similar transients as in Figures 4 and 5. The latter again speaks in favor of the UFIR filter, which has shorter transients.Figure 6
Vehicle tracking with temporary missing data in the east directionx by the UFIR filter, KF, and H∞ filter, all augmented with the prediction option. Tuned improperly, the H∞ filter becomes unstable and prone to divergence, as in Figure 3.


## 5.1. State-Space Model

To investigate the trade-off between the estimators, we will suppose that a vehicle is represented with two states in each directions and assign the state vector,K=4, as xn=x1nx2nx3nx4nT, where x1n=xn, x2n=x˙n, x3n=yn, and x4n=y˙n. Accordingly, the system matrix attains the form of(45)F=1τ000100001τ0001, where the sampling time is τ=1 s for the considered database. The GPS tracker provides measurements of the vehicle coordinates, x and y. Therefore, the measurement matrix is(46)H=10000010.ProvidedF and H, matrix Cm,s required by the UFIR filter to compute the initial values (40) and (41) for s=m+3 becomes(47)Cs,m=1-3τ00001-3τ1-2τ00001-2τ1-τ00001-τ10000010F-k.The only tuning factorNopt=5 required by the UFIR filter was found for k=0 by minimizing the derivative of the trace of the mean square value of the residual yn-Hx^n(N), as shown in [30]. Because Nopt depends on k, we will apply Nopt=5 in the worst case for the UFIR filter.Having no information about the process noise, we observe similar trajectories and estimate the average vehicle speed by about 10 m/s or 36 km/hour. Next, accepting the speed standard deviation of about 20%, we setσw2=2 m/s to the second state along each of the coordinates, ignore the unknown noise in the first state, σw1=0, and describe matrix Q as(48)Q=σw22τ2/2τ/200τ/210000τ2/2τ/200τ/21.The GPS standard positioning service provides navigation with an error of less than 15 meters with the probability of 95% in the 2-sigma sense. Referring to this value, we assign the standard deviation of the measurement noise in each direction asσv=15/4=3.75 m and obtain(49)R=σv200σv2.Because the above provided matricesQ and R are overestimated, we set Qˇ=Q and Rˇ=R for H∞. It is expected that the tuning factor θ will improve the performance of the H∞ filter by minimizing the MSE for the maximized errors. However, the ground truth is not available in tracking. Therefore, we will find θ for the measured trajectory and consider it as the best case for H∞, which is unfeasible.


## 5.2. Effect of Latency on the Estimation Accuracy

We start with learning the effect ofk on the estimation accuracy, which is illustrated in Figure 3 with the root MSEs (RMSEs) in the north direction (y-RMSE). The KF is self-tuned to k. Therefore, we consider its RMSE as a benchmark. A special feature of the KF is that the RMSE grows with k nonlinearly and faster than in the UFIR and H∞ filters.Figure 3
Effect of thek-step-lag on the y-RMSE of the UFIR filter, KF, and H∞ filter with different tunings.The UFIR filter produces a bit more errors than in the KF with smallk and lesser with larger k. A special feature is that the UFIR estimate is of low sensitivity to N, in which optimal value Nopt=5 holds for 0⩽k⩽4, increases to Nopt=6 for 5⩽k⩽6, and reaches Nopt=7 when 7⩽k⩽10. Of practical importance is that setting N optimally for each k does not improve the performance essentially against the worst case when Nopt=5 is set for all k.TheH∞ filter outperforms both the UFIR filter and KF, provided that θ is set properly for each lag k. However, this filter is highly sensitive to θ, in which optimal value θopt ranges from 1.8×10-2 for k=0 to 1.066×10-5 for k=10 in a nonlinear way. Unlike in the UFIR filter, a constant θ is unacceptable for all k. An example is given in Figure 3, where θopt=7.1×10-4 found for k=2 is applied in a wide range of k. As can be seen, it is only when k=2 that the H∞ filter improves the KF performance. For k<2, there is no improvement and, when k>2, the H∞ filter rapidly diverges.


## 5.3. Tracking over Data Delayed onk=3

We now suppose that data are transferred from a vehicle to a CS withk=3 and investigate tracking errors in the north and east directions. Filters will be tuned as follows: UFIR in the worst case of Nopt=5 being valid for k=0; KF as near optimal; and H∞ being in the best (unfeasible) case of known ground truth.


### 5.3.1. Tracking in the North Direction

The vehicle trajectory (measured and delayed) and estimates provided by the filters in the north direction are sketched in Figure4. As can be seen in Figure 4(a), the trajectory is nonlinear and the vehicle sometimes maneuvers rapidly. All filters produce consistent estimates with poorly distinguishable differences (Figure 4(a)). We therefore additionally show two parts of the trajectory on short-time spans of 0⩽n⩽200 in Figure 4(b) and 300⩽n⩽400 in Figure 4(c). Several observations can be made from Figure 4:(i)
All filters track well the trajectory when a vehicle travels with a near constant velocity in one direction, as on140⩽n⩽180 in Figure 4(b).(ii)
Filters temporarily lose an ability of tracking and go along the delayed data when a vehicle quickly changes the direction, as on340⩽n⩽343 in Figure 4(c).(iii)
Responding to fast maneuvers, all filters produce dynamic errors such that the UFIR filter comes up with larger excursions but shorter transients, KF with shorter excursions but longer transients, andH∞ filter demonstrates inbetween properties; see on 80⩽n⩽100 in Figure 4(b) and 344⩽n⩽360 in Figure 4(c).Figure 4
Tracking in the north directiony, m for k=3 with UFIR filter (Nopt=5), KF, and H∞ filter (θ=3.3×10-4): (a) full time scale, (b) 0⩽n⩽200, and (c) 300⩽n⩽400.
(a) (b) (c)Because errors are unacceptably large in all filters when they temporarily lose an ability of tracking, a key question arises of how fast each of them returns back to the normal mode. In this regard the UFIR filter looks better with its shortest transient.


### 5.3.2. Tracking in the East Direction

Tracking in the east direction (Figure5) does not reveal any essential features. The filters still track well the trajectory when a vehicle travels with a near constant velocity as, for example, in a span of 120⩽n⩽160 in Figure 5(b). Here, all filters also temporarily lose an ability of tracking and go along the delayed data when the trajectory quickly changes as, for example, in a span of 560⩽n⩽570 in Figure 5(c). Finally, the UFIR filter still demonstrates larger excursions but shorter transients, KF shorter excursions but longer transients, and H∞ filter is inbetween as, for example, in spans of 60⩽n⩽80 in Figure 5(b) and 570⩽n⩽580 in Figure 5(c).Figure 5
Tracking in the east directionx, m for k=3 with UFIR filter (Nopt=5), KF, and H∞ filter (θ=3.3×10-4): (a) full time scale, (b) 0⩽n⩽200, and (c) 500⩽n⩽600.
(a) (b) (c)


## 5.3.1. Tracking in the North Direction

The vehicle trajectory (measured and delayed) and estimates provided by the filters in the north direction are sketched in Figure4. As can be seen in Figure 4(a), the trajectory is nonlinear and the vehicle sometimes maneuvers rapidly. All filters produce consistent estimates with poorly distinguishable differences (Figure 4(a)). We therefore additionally show two parts of the trajectory on short-time spans of 0⩽n⩽200 in Figure 4(b) and 300⩽n⩽400 in Figure 4(c). Several observations can be made from Figure 4:(i)
All filters track well the trajectory when a vehicle travels with a near constant velocity in one direction, as on140⩽n⩽180 in Figure 4(b).(ii)
Filters temporarily lose an ability of tracking and go along the delayed data when a vehicle quickly changes the direction, as on340⩽n⩽343 in Figure 4(c).(iii)
Responding to fast maneuvers, all filters produce dynamic errors such that the UFIR filter comes up with larger excursions but shorter transients, KF with shorter excursions but longer transients, andH∞ filter demonstrates inbetween properties; see on 80⩽n⩽100 in Figure 4(b) and 344⩽n⩽360 in Figure 4(c).Figure 4
Tracking in the north directiony, m for k=3 with UFIR filter (Nopt=5), KF, and H∞ filter (θ=3.3×10-4): (a) full time scale, (b) 0⩽n⩽200, and (c) 300⩽n⩽400.
(a) (b) (c)Because errors are unacceptably large in all filters when they temporarily lose an ability of tracking, a key question arises of how fast each of them returns back to the normal mode. In this regard the UFIR filter looks better with its shortest transient.


## 5.3.2. Tracking in the East Direction

Tracking in the east direction (Figure5) does not reveal any essential features. The filters still track well the trajectory when a vehicle travels with a near constant velocity as, for example, in a span of 120⩽n⩽160 in Figure 5(b). Here, all filters also temporarily lose an ability of tracking and go along the delayed data when the trajectory quickly changes as, for example, in a span of 560⩽n⩽570 in Figure 5(c). Finally, the UFIR filter still demonstrates larger excursions but shorter transients, KF shorter excursions but longer transients, and H∞ filter is inbetween as, for example, in spans of 60⩽n⩽80 in Figure 5(b) and 570⩽n⩽580 in Figure 5(c).Figure 5
Tracking in the east directionx, m for k=3 with UFIR filter (Nopt=5), KF, and H∞ filter (θ=3.3×10-4): (a) full time scale, (b) 0⩽n⩽200, and (c) 500⩽n⩽600.
(a) (b) (c)


## 5.4. Tracking with Temporary Lost Data

We finally admit that some data points can be lost during the transmission and remove 5 data points atn=(615⋯620) s and 10 at n=(665⋯675) s as shown in Figure 6 for k=3. To predict lost data, we augment each algorithm with the prediction block as mentioned below (41) and run the filters. As can be seen, the filters act consistently with, however, some specifics. The estimates do not get away essentially from each other and the actual trajectory. However, when a vehicle maneuvers during the prediction, all filters diverge and return back to the actual trajectory with similar transients as in Figures 4 and 5. The latter again speaks in favor of the UFIR filter, which has shorter transients.Figure 6
Vehicle tracking with temporary missing data in the east directionx by the UFIR filter, KF, and H∞ filter, all augmented with the prediction option. Tuned improperly, the H∞ filter becomes unstable and prone to divergence, as in Figure 3.


## 6. Conclusions

The UFIR filter developed in this paper for GPS-based vehicle tracking over WSNs with time-stamped discretely delayed and missing data has demonstrated better performance than the KF andH∞ filter. The main benefits of using the UFIR filter are that it (1) does not require any information about noise and initial conditions, (2) becomes blind on given horizons, and (3) has shorter transients. The latter can be considered as an important practical advantage in all situations when the trajectory changes rapidly and estimators temporarily loses an ability of tracking. Applications to GPS-based vehicle tracking with known discretely delayed and missed data have proved a better performance of the UFIR filter.



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*Source: 7456010-2018-05-08.xml*