Constrained Weighted Least Squares Research Articles

Source Localization by Frequency Measurements in Unknown Signal Propagation Speed Environments

Signal propagation speed may not be available in practical localization scenarios. This paper addresses both the moving and stationary source localization problems by using frequency measurements when the signal propagation speed is unavailable. The task is fulfilled by exploiting the Doppler frequency shift coming from the relative motion between the source and a network of mobile sensors. Owing to the complicated nonlinear measurement model and the absence of the knowledge about signal propagation speed, it is difficult to conduct localization. To address this problem, we first transform the measurement model to a tractable form, from which to formulate a constrained weighted least squares (CWLS) problem. Afterwards, we propose two methods to solve the non-convex CWLS problem. The first method relaxes the CWLS problem into a semidefinite programming (SDP) problem by applying semidefinite relaxation (SDR). The second method adopts the alternating estimation procedure by utilizing the prior information about the nominal value of the signal propagation speed. Two subproblems are formulated and solved in an alternate manner, one in estimating the source position and velocity by applying SDR and the other the signal propagation speed by an explicit expression. Furthermore, we conduct the mean square error (MSE) analysis to show that the CWLS solution is able to reach the Cramér-Rao lower bound (CRLB) performance. Simulation results validate the expected performance of the proposed methods.

Source Localization Based on Hybrid AOA, TDOA, and RSS Measurements

In this paper, a constrained weighted least squares (CWLS) problem with two quadratic constraints for hybrid AOA, TDOA, and RSS localization is formulated. The Lagrange multiplier method cannot be efficiently used in this situation, so we provide four methods to solve the CWLS problem: 1) weighted least squares (WLS); 2) iterative constrained weighted least squares (ICWLS); 3) Lagrange multiplier method combined with Newton’s method (LMM+NM); 4) semidefinite programming (SDP). Furthermore, in order to obtain a good balance between accuracy and computational burden, we develop a fixed point iteration (FPI)-based algorithm that is derived from an approximate maximum likelihood estimator (MLE). Simulation results show that the accuracy of the proposed FPI method is better than the other four methods; besides, its computational burden is one in ten of the SDP algorithm.

Motion Parameter Estimation for Mobile Sources Using Semidefinite Programming

In this paper, we address the motion parameter estimation problem using time difference of arrival (TDOA) measurements, where a mobile source starts from an initial position with constant velocity. The problem is formulated as three non-convex constrained weighted least squares (CWLS) problems. Owing to their non-convex nature, local convergence may occur when solving them by an iterative algorithm, implying good estimates are not guaranteed. We propose to solve these CWLS problems by applying semidefinite relaxation (SDR) to generate three different semidefinite programming (SDP) problems, namely, the motion unconstrained semidefinite programming (MU-SDP), motion constrained semidefinite programming (MC-SDP), and motion parameter direct semidefinite programming (MPD-SDP). The MU-SDP, MC-SDP, and MPD-SDP methods are then extended to the scenario of unknown propagation speed (PS), which is common in underwater and underground localizations. Specifically, for this scenario, the two-step MU-SDP, two-step MC-SDP, and two-step MPD-SDP methods are designed to keep the relaxed SDP problems tight by introducing the penalty function. Simulation results show that MU-SDP performs worse than MC-SDP and MPD-SDP owing to the ignorance of the motion equation, and MC-SDP and MPD-SDP are able to reach the Cramér-Rao lower bound (CRLB) accuracy.

Reducing Bias for Multistatic Localization of a Moving Object by Transmitter At Unknown Position

Multistatic localization plays an important role in object localization. In this paper, we address the problem of multistatic localization of a moving object in the absence of transmitter position when the transmitter is not synchronized with the receivers. We propose to jointly estimate the unknowns, including the object position and velocity, the transmitter position, and the clock and frequency offsets, using both time delay and Doppler frequency shift measurements. To this end, we first formulate a constrained weighted least squares (CWLS) minimization problem, whose solution is found to have significant bias caused by approximations in transforming the model equations and solving the problem. To reduce the bias, we further formulate a non-convex bias reduced CWLS (BR-CWLS) problem by imposing a quadratic constraint, which is constructed by considering the second-order noise and errors of the matrices and vectors involved in the CWLS problem. One particular aspect of the proposed BR-CWLS method is that the errors in approximating the weighting matrix are taken into consideration for bias reduction. The non-convex BR-CWLS problem is solved by applying the semidefinite relaxation technique to relax it as a convex semidefinite program. In addition, we show through the mean square error (MSE) analysis that the performance of the BR-CWLS solution can approach the Cramer-Rao lower bound performance when the noise is not significant. We also derive the theoretical bias expression for evaluating the amount of bias. Simulation results demonstrate the good performance of the proposed method in terms of both MSE and bias.

Multistatic Localization by Differential Time Delays and Time Differences of Arrival in the Absence of Transmitter Position

We can only extract the differential time delay (DTD) measurements between the direct and reflected paths in multistatic localization when there is no synchronization in time between the transmitter and receiver and among the receivers. This paper first addresses the problem of multistatic localization of a fixed object when the transmitter position is not available by using the DTD measurements. We propose a two-step optimization method for jointly estimating the object and transmitter positions. In the first step, we formulate a non-convex constrained weighted least squares (CWLS) problem by transforming the DTD measurement model and introducing nuisance variables. Such a non-convex CWLS problem is then relaxed to a tractable convex semidefinite programming (SDP) problem by applying semidefinite relaxation. In the second step, the error coming from relaxation and approximation in the SDP solution is reduced iteratively through solving a generalized trust region subproblem (GTRS) in each iteration. If the receivers are synchronized such that the time difference of arrival (TDOA) measurements can be acquired in addition to DTD, we formulate a different CWLS problem by utilizing both DTD and TDOA measurements, which is solved by convex relaxation as well. The relaxed SDP problem can achieve the optimal solution of the CWLS problem, and further refinement is no longer needed. We conduct the mean square error (MSE) analysis to validate that both proposed methods are able to achieve the Cramer-Rao Lower Bound (CRLB) performance under small Gaussian noise, which is also validated by simulations.

A Semidefinite Relaxation Solution for Time Delay and Doppler Shift Localization Considering Sensor Location Errors and Its Bias Reduction Scheme

This article develops a robust source localization method using time delay and Doppler shift measurements, where the sensor motion effect accompanied by sensor location errors cannot be ignored. We begin by transforming the time delay and Doppler shift measurement models into a series of nonlinear equations that take sensor location errors into account, and then construct a constrained weighted least squares (CWLS) problem based on these equations. Because of the nonconvex nature of the problem, we relax it into a semidefinite programming (SDP) problem via convex relaxation and further propose a scheme to eliminate the influence of the additional estimation bias caused by the approximation applied in the transformation of measurement models. To perform the bias reduction, we derive the theoretical expression of the solution bias and then subtract it to obtain a bias-reduced solution. We also derive the closed-form expression of the hybrid Cramer–Rao lower bound (HCRLB) as the performance benchmark. Theoretical analysis and simulation results demonstrate that the mean-square error (MSE) performance of the proposed method can achieve the HCRLB accuracy and the bias can be significantly reduced with bias reduction.

A Convex Relaxation Algorithm for Source Localization Considering Sensor Motion in Wireless Sensor Networks

In this letter, by using the effect of sensor motion on time delay and Doppler shift measurements, a semi-definite programming (SDP) solution is developed via convex relaxation to determine the unknown source location. A novel constrained weighted least squares (CWLS) problem is formulated according to the non-recursive measurement models without any approximation. After that, semi-definite relaxation (SDR) is performed to transform the CWLS problem into a convex programming problem, which can guarantee the global optimal solution. Simulation results show that the proposed method can provide a significant reduction in mean-square error (MSE) compared with the existing methods.

Passive Target Localization Problem Based on Improved Hybrid Adaptive Differential Evolution and Nelder-Mead Algorithm

This paper considers a passive target localization problem in Wireless Sensor Networks (WSNs) using the noisy time of arrival (TOA) measurements, obtained from multiple receivers and a single transmitter. The objective function is formulated as a maximum likelihood (ML) estimation problem under the Gaussian noise assumption. Consequently, the objective function of the ML estimator is a highly nonlinear and nonconvex function, where conventional optimization methods are not suitable for this type of problem. Hence, an improved algorithm based on the hybridization of an adaptive differential evolution (ADE) and Nelder-Mead (NM) algorithms, named HADENM, is proposed to find the estimated position of a passive target. In this paper, the control parameters of the ADE algorithm are adaptively updated during the evolution process. In addition, an adaptive adjustment parameter is designed to provide a balance between the global exploration and the local exploitation abilities. Furthermore, the exploitation is strengthened using the NM method by improving the accuracy of the best solution obtained from the ADE algorithm. Statistical analysis has been conducted, to evaluate the benefits of the proposed modifications on the optimization performance of the HADENM algorithm. The comparison results between HADENM algorithm and its versions indicate that the modifications proposed in this paper can improve the overall optimization performance. Furthermore, the simulation shows that the proposed HADENM algorithm can attain the Cramer-Rao lower bound (CRLB) and outperforms the constrained weighted least squares (CWLS) and differential evolution (DE) algorithms. The obtained results demonstrate the high accuracy and robustness of the proposed algorithm for solving the passive target localization problem for a wide range of measurement noise levels.

Time Difference of Arrival (TDoA) Localization Combining Weighted Least Squares and Firefly Algorithm

Time difference of arrival (TDoA) based on a group of sensor nodes with known locations has been widely used to locate targets. Two-step weighted least squares (TSWLS), constrained weighted least squares (CWLS), and Newton–Raphson (NR) iteration are commonly used passive location methods, among which the initial position is needed and the complexity is high. This paper proposes a hybrid firefly algorithm (hybrid-FA) method, combining the weighted least squares (WLS) algorithm and FA, which can reduce computation as well as achieve high accuracy. The WLS algorithm is performed first, the result of which is used to restrict the search region for the FA method. Simulations showed that the hybrid-FA method required far fewer iterations than the FA method alone to achieve the same accuracy. Additionally, two experiments were conducted to compare the results of hybrid-FA with other methods. The findings indicated that the root-mean-square error (RMSE) and mean distance error of the hybrid-FA method were lower than that of the NR, TSWLS, and genetic algorithm (GA). On the whole, the hybrid-FA outperformed the NR, TSWLS, and GA for TDoA measurement.

Multi-source passive localization via multiple unmanned aerial vehicles

Passive source localization via multiple unmanned aerial vehicles (UAVs) is a key technology for the practical application of military reconnaissance. Previous studies primarily focused on the localization of a single source. This study explores the multi-source passive localization problem using time difference of arrival (TDOA) measurements. We formulate the localization problem as a constrained weighted least squares problem. The formulation is an indefinite quadratically constrained quadratic programming problem, which is non-convex and NP-hard. To obtain approximate programming with linear constraints, an iterative constrained weighted least squares (CWLS) algorithm is proposed to perform a linearization procedure on the quadratic equality constraints. Theoretical analysis reveals that the proposed algorithm, if converges, can lead to a global optimal solution of the formulated problem. The results of the Monte Carlo experiment indicate that the proposed algorithm quickly converges in most situations and offers better localization accuracy compared with the previous two-step weighted least squares method.

Sensor Network-Based Rigid Body Localization via Semi-Definite Relaxation Using Arrival Time and Doppler Measurements

This paper addresses the rigid body localization problem using a convex optimization approach. We propose a semi-definite relaxation (SDR) method for locating a stationary rigid body using arrival time measurements, and extend it for moving rigid body using both arrival time and Doppler measurements. Localization of a stationary (moving) rigid body involves not only the position (and velocity) but also the rotation angles (and angular velocity), making it a challenging optimization problem with nonlinear constraints. We approximate the maximum likelihood problem with a constrained weighted least-squares (CWLS) minimization and apply SDR to obtain a coarse estimate. The orthogonalization and refinement procedures are followed next to recover the performance loss caused by relaxation and approximation. It is shown analytically that the CWLS solution can achieve the Cramer–Rao lower bound accuracy for Gaussian noise when the noise level is not significant. The simulations show that the proposed method achieves better accuracy than the previously developed methods.

Iterative Target Localization in Distributed MIMO Radar From Bistatic Range Measurements

This letter discusses the problem of determining the target position from bistatic range measurements in a multiple-input multiple-output radar system with widely separated antennas. By introducing nuisance parameters and taking the relationship between these parameters and the unknown target position into consideration, the problem is formulated as a constrained weighted least squares (CWLS) problem, which is a quadratically constrained quadratic programming problem. Since the constraints are nonconvex, finding the global solution is a difficult task. To solve this problem, an iterative CWLS method is developed in this letter. We iteratively approximate each nonconvex constraint as a linear constraint to convert the problem as a linearly constrained quadratic programming problem and then solve the problem. Numerical simulations are included to support and corroborate the theoretical developments.

Iterative Constrained Weighted Least Squares Source Localization Using TDOA and FDOA Measurements

This paper investigates the constrained weighted least squares (CWLS) source localization problem by using time difference of arrival and frequency difference of arrival measurements. The problem can be formulated as a quadratic programming with two indefinite quadratic equality constraints, which is nonconvex and NP-hard. Moreover, the weighting matrix is coupled with the unknown source position and velocity. We propose an iterative CWLS method that can efficiently solve this problem. It iteratively performs a linearization procedure on the quadratic equality constraints to obtain an approximate programming with linear constraints, which can be analytically solved, and the weighting matrix is updated in each iteration. Theoretical analysis reveals that the proposed method, if converges, can lead to the global optimal solution of the formulated problem that reaches the CRLB accuracy under mild assumptions on the measurement noises. The Monte Carlo simulation results indicate that the percentage of convergence within $\mathbf 20$ iterations is more than $\mathbf 96\%$ , and the localization accuracy is significantly improved over the previous methods with less computation time requirement. Moreover, it is found from simulations that the iterative CWLS method retains acceptable performance even under the ill-conditioned situation when the sensor geometry is not desirable.

Tensor estimation for double-pulsed diffusional kurtosis imaging.

Double-pulsed diffusional kurtosis imaging (DP-DKI) represents the double diffusion encoding (DDE) MRI signal in terms of six-dimensional (6D) diffusion and kurtosis tensors. Here a method for estimating these tensors from experimental data is described. A standard numerical algorithm for tensor estimation from conventional (i.e. single diffusion encoding) diffusional kurtosis imaging (DKI) data is generalized to DP-DKI. This algorithm is based on a weighted least squares (WLS) fit of the signal model to the data combined with constraints designed to minimize unphysical parameter estimates. The numerical algorithm then takes the form of a quadratic programming problem. The principal change required to adapt the conventional DKI fitting algorithm to DP-DKI is replacing the three-dimensional diffusion and kurtosis tensors with the 6D tensors needed for DP-DKI. In this way, the 6D diffusion and kurtosis tensors for DP-DKI can be conveniently estimated from DDE data by using constrained WLS, providing a practical means for condensing DDE measurements into well-defined mathematical constructs that may be useful for interpreting and applying DDE MRI. Data from healthy volunteers for brain are used to demonstrate the DP-DKI tensor estimation algorithm. In particular, representative parametric maps of selected tensor-derived rotational invariants are presented.

Constrained weighted least Squares location algorithm using Received Signal Strength measurements

Determine the location of a target has gained considerable interest over the past few years. The Received Signal Strength (RSS) measurements and Differential RSS (DRSS) measurements can be converted to distance or distance ratio estimates for constructing a set of linear equations. Based on these linear equations, a constrained weighted least Squares (CWLS) algorithm for target localization is derived. In addition, an iterative technique based on Newton's method is utilized to give a solution. The covariance and bias of the CWLS algorithm is derived using perturbation analysis. Simulation shows that the proposed estimator achieves better performance than existing algorithms with reasonable complexity.

Joint Frequency and Phasor Estimation Under the KCL Constraint

In this letter, we consider the problem of joint off-nominal frequency and phasor estimation that incorporates Kirchhoff's Current Law (KCL) as a constraint. We develop the constrained maximum likelihood (CML) and constrained weighted least-squares (CWLS) estimators for this problem and derive the corresponding constrained Cramer–Rao bound. The KCL constraint is shown to behave as a noise cancellation factor for the phasors estimation. We show that the KCL CML is based on the classical periodogram subtracting the average current periodogram. The results indicate significant performance improvement compared to the unconstrained maximum likelihood (ML) and unconstrained weighted least-squares (WLS).

A new constrained weighted least squares algorithm for TDOA-based localization

The linear least squares (LLS) technique is widely used in time-difference-of-arrival based positioning because of its computational efficiency. Two-step weighted least squares (2WLS) and constrained weighted least squares (CWLS) algorithms are two common LLS schemes where an additional variable is introduced to obtain linear equations. However, they both have the same measurement matrix that becomes ill-conditioned when the sensor geometry is a uniform circular array and the source is close to the array center. In this paper, a new CWLS estimator is proposed to circumvent this problem. The main strategy is to separate the source coordinates and the additional variable to different sides of the linear equations where the latter is first solved via a quadratic equation. In doing so, the matrix to be inverted has a smaller condition number than that of the conventional LLS approach. The performance of the proposed method is analyzed in the presence of zero-mean white Gaussian disturbances. Numerical examples are also included to evaluate its localization accuracy by comparing with the existing 2WLS and CWLS algorithms as well as the Cramér–Rao lower bound.

Optimal Reconfiguration Control of the Yaw Stability of the Tractor-Semitrailer Vehicle

An optimal reconfiguration control scheme based on control allocation (CA) is proposed to stabilize the yaw dynamics of the tractor-semitrailer vehicle. The proposed control scheme is a two-level structure consisting of an upper level of sliding mode yaw moment controller (SMYC) and a lower optimal brake force distributor (BFD). The upper SMYC is designed to follow the tractor yaw rate and the combination of the hitch angle and trailer slip angle and outputs the corrective yaw moment, respectively, for the tractor and the trailer. The optimal brake force allocation and reconfigurable control problem is transformed to a problem of error minimization and control minimization combination formulated by constrained weighted least squares (CWLS) optimization and further solved with active set (AS) algorithm. Simulation results reveal that the CA technique-based optimal reconfigurable control is rather effective for the tractor-semitrailer vehicle to enhance the yaw stability performance and the reliability in case of actuator failure thanks to the multiple-axle structure enriching the alternatives of possible actuator combinations in CA optimization.

An Efficient Constrained Weighted Least Squares Algorithm for Moving Source Location Using TDOA and FDOA Measurements

In this paper, by utilizing the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements of a signal received at a number of receivers, an efficient constrained weighted least-squares (CWLS) algorithm for estimating the position and velocity of a moving source is proposed, which exploits the known relation between the intermediate variable and the source location coordinates explicitly. On basis of Newton's method, a numerical iterative solution can be obtained allowing real-time implementation and ensuring global convergence. Simulation results show that the proposed estimator achieves remarkably better performance than the two-step weighted least squares (WLS) approach, which makes the Cramr-Rao lower bound (CRLB) at a sufficiently high noise level before the threshold effect occurs.

A Constrained Least Squares Approach to Mobile Positioning: Algorithms and Optimality

The problem of locating a mobile terminal has received significant attention in the field of wireless communications. Time-of-arrival (TOA), received signal strength (RSS), time-difference-of-arrival (TDOA), and angle-of-arrival (AOA) are commonly used measurements for estimating the position of the mobile station. In this paper, we present a constrained weighted least squares (CWLS) mobile positioning approach that encompasses all the above described measurement cases. The advantages of CWLS include performance optimality and capability of extension to hybrid measurement cases (e.g., mobile positioning using TDOA and AOA measurements jointly). Assuming zero-mean uncorrelated measurement errors, we show by mean and variance analysis that all the developed CWLS location estimators achieve zero bias and the Cramer-Rao lower bound approximately when measurement error variances are small. The asymptotic optimum performance is also confirmed by simulation results.