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Distributed Target Tracking based on Localization with Linear Time-Difference-of-Arrival Measurements: A Delay-Tolerant Networked Estimation Approach

Mohammadreza Doostmohammadian, Themistoklis Charalambous

TL;DR

This work tackles distributed target tracking using time-difference-of-arrival (TDOA) measurements across a network of static sensors, without requiring local observability at every node. It introduces a linear, time-invariant TDOA measurement model with a fixed output matrix $H_i$, and an offline block-diagonal gain $K$ enabling a single time-scale estimator whose stability is guaranteed when $(W \otimes F, D_H)$ is observable; it also proves delay-tolerance under heterogeneous fixed delays using $F^{\overline{\tau}+1}$ and a common $K$. The authors extend the framework to $q$-redundant, survivable networks and discuss joint distributed fault detection, providing a comprehensive delay-aware, decentralized solution with reduced communication compared to double-time-scale methods. Through simulations, the approach demonstrates stable, bounded MSEE under delays and link failures, and shows advantages over DTS in terms of communication overhead while maintaining tracking accuracy. Overall, the paper presents a practical, scalable, and robust distributed tracking paradigm suitable for static sensor networks and adaptable to extensions for mobile sensor formations.

Abstract

This paper considers target tracking based on a beacon signal's time-difference-of-arrival (TDOA) to a group of cooperating sensors. The sensors receive a reflected signal from the target where the time-of-arrival (TOA) renders the distance information. The existing approaches include: (i) classic centralized solutions which gather and process the target data at a central unit, (ii) distributed solutions which assume that the target data is observable in the dense neighborhood of each sensor (to be filtered locally), and (iii) double time-scale distributed methods with high rates of communication/consensus over the network. This work, in order to reduce the network connectivity in (i)-(ii) and communication rate in (iii), proposes a distributed single time-scale technique, which can also handle heterogeneous constant data-exchange delays over the static sensor network. This work assumes only distributed observability (in contrast to local observability in some existing works categorized in (ii)), i.e., the target is observable globally over a (strongly) connected network. The (strong) connectivity further allows for survivable network and $q$-redundant observer design. Each sensor locally shares information and processes the received data in its immediate neighborhood via local linear-matrix-inequalities (LMI) feedback gains to ensure tracking error stability. The same gain matrix works in the presence of heterogeneous delays with no need of redesigning algorithms. Since most existing distributed estimation scenarios are linear (based on consensus), many works use linearization of the existing nonlinear TDOA measurement models where the output matrix is a function of the target position.

Distributed Target Tracking based on Localization with Linear Time-Difference-of-Arrival Measurements: A Delay-Tolerant Networked Estimation Approach

TL;DR

This work tackles distributed target tracking using time-difference-of-arrival (TDOA) measurements across a network of static sensors, without requiring local observability at every node. It introduces a linear, time-invariant TDOA measurement model with a fixed output matrix , and an offline block-diagonal gain enabling a single time-scale estimator whose stability is guaranteed when is observable; it also proves delay-tolerance under heterogeneous fixed delays using and a common . The authors extend the framework to -redundant, survivable networks and discuss joint distributed fault detection, providing a comprehensive delay-aware, decentralized solution with reduced communication compared to double-time-scale methods. Through simulations, the approach demonstrates stable, bounded MSEE under delays and link failures, and shows advantages over DTS in terms of communication overhead while maintaining tracking accuracy. Overall, the paper presents a practical, scalable, and robust distributed tracking paradigm suitable for static sensor networks and adaptable to extensions for mobile sensor formations.

Abstract

This paper considers target tracking based on a beacon signal's time-difference-of-arrival (TDOA) to a group of cooperating sensors. The sensors receive a reflected signal from the target where the time-of-arrival (TOA) renders the distance information. The existing approaches include: (i) classic centralized solutions which gather and process the target data at a central unit, (ii) distributed solutions which assume that the target data is observable in the dense neighborhood of each sensor (to be filtered locally), and (iii) double time-scale distributed methods with high rates of communication/consensus over the network. This work, in order to reduce the network connectivity in (i)-(ii) and communication rate in (iii), proposes a distributed single time-scale technique, which can also handle heterogeneous constant data-exchange delays over the static sensor network. This work assumes only distributed observability (in contrast to local observability in some existing works categorized in (ii)), i.e., the target is observable globally over a (strongly) connected network. The (strong) connectivity further allows for survivable network and -redundant observer design. Each sensor locally shares information and processes the received data in its immediate neighborhood via local linear-matrix-inequalities (LMI) feedback gains to ensure tracking error stability. The same gain matrix works in the presence of heterogeneous delays with no need of redesigning algorithms. Since most existing distributed estimation scenarios are linear (based on consensus), many works use linearization of the existing nonlinear TDOA measurement models where the output matrix is a function of the target position.

Paper Structure

This paper contains 19 sections, 7 theorems, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Framework
  3. Problem statement
  4. The target model
  5. The Proposed Sensor Measurement Model
  6. Single Time-Scale Networked Estimation
  7. The proposed estimation protocol
  8. Structural design of the sensor network
  9. Tracking in Presence of Heterogeneous Delays
  10. More Discussions and Auxiliary Results
  11. $q$-Redundant distributed observer design
  12. Brief overview on adding local fault detectors
  13. Some relevant remarks
  14. Simulation Results
  15. Linear versus linearized TDOA models: centralized case
  16. ...and 4 more sections

Key Result

Lemma 1

Let $\mathbf{e}_{i}(k)=\mathbf{x}(k)-\widehat{\mathbf{x}}_i(k)$ denote the tracking error at sensor $i$ and $\mathbf{e}(k)=(\mathbf{e}_{1}(k); \dots;\mathbf{e}_{n}(k))$ as the collective error vector. The error dynamics of the networked estimator eq_p-eq_m is, where $\otimes$ denotes the Kronecker matrix product, $D_H := \hbox{blkdiag}[H_i^\top H_i]$ and $K:= \hbox{blkdiag}[K_{i}]$ are defined as

Figures (4)

  • Figure 1: Two types of consensus-based networked estimation mechanisms: (Left) double time-scale (DTS), and (Right) single time-scale (STS). The DTS scenario imposes many (fast) iterations of consensus and communications (in general, more than the network diameter $d_n$) to perform estimation and address observability. In contrast, in the STS method, only one round of inter-sensor message-exchange is done per observation/system epoch.
  • Figure 2: This figure presents the target tracking setup in this paper: a group of geographically distributed static sensors (radars) receive a beacon signal from a target (drone) and locally (individually) track the target via distributed estimation and localized fault detection techniques. The measurements are based on the time-difference-of-arrival (TDOA) measurements described via Eq. \ref{['eq_h_simp']} and Eq. \ref{['eq_tdoa']}.
  • Figure 3: This figure compares the MSEE performance of the linear measurement model \ref{['eq_h_simp']}-\ref{['eq_tdoa_new']} versus the linearized model \ref{['eq_tdoa']}-\ref{['eq_hij']} used in ennasr2016distributedennasr2020time for different measurement and process noise variances. Note that, for the nonlinear case, the linearized output matrix $\overline{H}_i$ is a function of the target position itself. For the simulation, both the true target position and estimated target position are considered in the output matrix for the sake of simulation.
  • Figure 4: This figure shows the performance of the STS protocol \ref{['eq_p']}-\ref{['eq_m']} for tracking the NCV target compared to the DTS protocol in he2020secure. (LeftTop) The position of sensors, their cyclic communication network (red-colored), and the initial position of the target (black cross), (MidTop) the target path and final position (black-colored) are shown over $4000$ time-steps. (RightTop) The MSEE performance averaged at all sensors over $20$ MC trials is shown, which is steady-state stable and implies observable estimation. (RightTop) The position and velocity errors at (an example) sensor $4$ are unbiased in steady-state. (LeftMid) Target tracking subject to heterogeneous delays at different links with max delays $\overline{\tau} = 4,8$: MSEE MC simulation averaged over $10$ trials for all sensor states and (MidMid) state errors at a randomly chosen sensor with $\overline{\tau} = 4$. The noise variances are different for this example. (RightMid) The (averaged) MSEE over the reduced sensor-network after the removal of one link is steady-state stable and thus distributed observability is preserved after link failure. (Bottom) The MSEE performance of the DTS protocol with $L$ additional epochs of averaging (consensus) and communication between every two consecutive time-steps for two different $\beta$ values are presented. For this DTS simulation, more measurements are considered at every sensor (since the system is marginally stable he2020secure) such that the target is locally observable. From the figure, as the target moves away from the sensors the MSEE performance degrades and more consensus epochs (larger $L$) might be needed.

Theorems & Definitions (23)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Remark 4
  • Lemma 4
  • ...and 13 more