Nonlinear Measurement Equations Research Articles

Improved-RSSI-based indoor localization by using pseudo-linear solution with machine learning algorithms

With the rapid advancement of the Internet of Things and the popularization of mobile Internet-based applications, the location-based service (LBS) has attracted much attention from commercial developers and researchers. Received signal strength indicator (RSSI)-based indoor localization technology has irreplaceable advantages for many LBS applications. However, due to multipath fading, noise, and the limited dynamic range of the RSSI measurements, precise localization based on a path-loss model and multiliterate becomes highly challenging. Therefore, this study proposes a machine learning (ML)-based improved RSSI-based indoor localization approach in which RSSI data is first augmented and then classified using ML algorithms. In addition, we implement an experimental testbed to collect the RSSI value based on Wi-Fi using various reference and target nodes. The received RSSI measurements undergo pre-processing using pseudo-linear solution techniques for closed-form solutions, approximating the original system of nonlinear RSSI measurement equations with a system of linear equations. Finally, the RSSI measurement are trained using ML models such as linear regression, polynomial regression, support vector regression, random forest regression, and decision tree regression. Consequently, the experimental results express in terms of root mean square error and coefficient of determinant compared with various machine learning models with hyper-parameter tuning.

Cyber-Physical System-Based Path Tracking Control of Autonomous Vehicles Under Cyber-Attacks

As one of the typical applications of cyber-physical system (CPS), autonomous vehicles (AVs) are vulnerable to malicious disturbance from cyber-attacks while tracking the desired path. This paper focuses on CPS-based path tracking control problem of AVs under cyber-attacks. Firstly, the nonlinear state and measurement equations of AVs under cyber-attacks are established based on vehicle dynamics model. Secondly, to improve the robustness of AVs against cyber-attacks, sensor redundancy is introduced. A cyber-attack detection method is designed by using extended Kalman filter (EKF), and the computational complexity of the cyber-attack detection method is analyzed. In addition, sensor switching rules are developed to isolate the disturbance of cyber-attacks. Then, the control problem of the AVs is formulated based on model predictive control (MPC). Input-to-state stability (ISS) of the control system under cyber-attacks is established, and a link between the tolerable attack intensity and the detection thresholds is clearly revealed. Finally, the simulation results demonstrate the effectiveness of the proposed control strategy.

A Nonlinear Filter by Pseudo-Formal Linearization for Nonlinear Time-Varying Systems

This paper is concerned with a nonlinear filter for a general class of nonlinear time-varying systems, by making use of a pseudo-formal linearization method and Kalman filter approach. The given nonlinear time-varying dynamic and measurement equations are piecewisely linearized by the formal linearization method using Taylor expansion so that Kalman filter can be applied for each linearized time-varying equations. Then the resulting filters are smoothly united into a single nonlinear filter by an automatic choosing function. The efficiency of this filter is shown through numerical examples.

TDOA –FDOA method for tracking of a moving target in a distributed sensor scenario

Tracking of an underwater target in ocean environment is very important in both military and civilian applications. This paper focuses on the localization and tracking of a moving target by a passive multi-static sonar system using TOA, Time Difference and Frequency Difference of Arrival measurements. An algorithm is developed for an acoustic sensor distributed network of five sensors and an acoustic moving source. TDOAFDOA method is used to track a moving source usingTime Difference of Arrival (TDOA) and Frequency Difference of Arrival (FDOA) measurements, as a closed form solution. The highly nonlinear measurement equations of TDOA and FDOA are first converted into pseudo linear equations. This is achieved by introducing extra parameters. These equations are solved using a weighted least squares (WLS) method. Five distributed stationary receivers are used to track a moving acoustic target. The moving target parameters estimated from the above WLS method are improved by an Unscented Kalman Filter. The mean square errors of these estimates are compared and the results are analyzed.

Passive UAV Tracking in Wireless Networks

Techniques for high accuracy tracking of the 3-D velocity, position, and cellular carrier frequency bias of unmanned aerial vehicles (UAVs) that communicate using a cellular network are presented. The tracking algorithm does <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">only</i> rely on asynchronous measurements of Doppler shift in a minimum of four base station sites. This number can be reduced to three, by combination with a round-trip time measurement from one site. The tracking is therefore independent of any capability of the UAV transceiver, like the global positioning system that may be disabled in the UAV by the pilot. Another significant advantage is that high precision time synchronization between base stations is not needed. The algorithm exploits interacting multiple model filtering, with the nonlinear measurement equations handled by extended Kalman filters. The low computational complexity, low sampling rate, and the uplink Doppler shift measurement resources enable joint tracking of several UAVs per cell. This makes the tracking technique relevant for supervision of UAV traffic using cellular connectivity, for example for border surveillance and surveillance of sensitive areas like airports where illegal flying may be extremely dangerous and costly. Theoretically, an observability analysis treats measurement configurations based on different combinations of Doppler shift and round-trip time. Simulations are used to illustrate the performance and to assess the initialization properties.

Particle filter underwater terrain-aided navigation based on gradient fitting

Inertial navigation systems (INSs) are widely used for autonomous underwater vehicle navigation, with excellent short-term precision. However, the positioning error of an INS accumulates with time. It is imperative to adopt other approaches to mitigate drift errors, especially for long-term sailing tasks. To that end, underwater terrain-based navigation (TBN) is effective in bounding the drift errors. In particular, the particle filtering (PF) method is extensively employed in TBN to tackle the highly nonlinear measurement equation. In our previous work, the statistical properties of the digital terrain model gradient were exploited and we proposed the use of three probability density functions to characterize the normalized data, i.e. Gaussian, gamma and Weibull distributions. In this paper, the likelihood was modified according to the gradient fitting results, i.e. an optimal distribution selection. Moreover, to prevent the measurement data with large gradients from generating particles with lower weights, a pre-screen procedure was proposed to stabilize particle filter sampling. As shown and demonstrated in simulations, our proposed improved PF method outperforms the comparative ones in terms of root mean square error and standard deviation, as well as accuracy, especially for long-term sailing tasks. In terms of computational cost, a smaller number of measurement data are employed and the proposed method is faster than the standard PF method.

A Nonlinear Filter with Constant‐Gains Using a Pseudo‐Formal Linearization Method Based on Chebyshev Expansion

This paper deals with a design of a nonlinear filter using a pseudo‐formal linearization method based on Chebyshev expansion with constant filter gains. The given nonlinear dynamic and measurement equations are piecewisely linearized by a formal linearization method using Chebyshev expansion, therefore the steady‐state Kalman filter can be applied for each linearized system. Then the resulting filters are smoothly united into a single nonlinear filter by an automatic choosing function. We demonstrate the proposed method in nonlinear systems to verify the effectiveness. © 2022 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Designing a Nonlinear Filter of Constant-Gains Type Using Pseudo-Formal Linearization Method

This paper is concerned with the designing a nonlinear filter making use of a pseudo-formal linearization method and Kalman filter approach of constant-gains type for nonlinear systems. We piecewisely linearize the given nonlinear dynamic and measurement equations by a formal linearization method and design linear filters. They are smoothly united into a single nonlinear filter. The efficacy of this filter is shown through numerical examples.

Rapid Reduction Dimensionality Transfer Alignment Method Based on CKF for SINS of Ship-borne Aircraft with GNSS Assistance

Abstract Aiming at the problem of rapid transfer alignment of inertial navigation system in shipborne environment with large azimuth misalignment angle, a dimension reduction fast transfer alignment scheme of shipborne aircraft inertial navigation system based on CKF algorithm is designed. Firstly, the nonlinear attitude error equation, velocity error equation and position error equation of the Global Navigation Satellite System (GNSS) assisted strapdown inertial navigation system in the case of large azimuth misalignment are analyzed. Secondly, the nonlinear dimensionality reduction rapid transfer alignment state equation and measurement equation are derived. Finally, the CKF algorithm is simulated and verified, and compared with EKF algorithm and UKF algorithm. The simulation results show that the reduced dimension fast transfer alignment method of shipborne aircraft inertial navigation system based on CKF algorithm can complete the estimation of attitude error angle in 80s, with horizontal estimation error of 1’ and azimuth error of 4’, and the estimation accuracy is equivalent to EKF and UKF, with less calculation and better rapidity.

Robust strong tracking unscented Kalman filter for non‐linear systems with unknown inputs

This paper proposes a state estimation approach ‘robust strong tracking unscented Kalman filter with unknown inputs’ that can be applied to non-linear systems with unknown inputs. Specifically, the non-linear state and measurement equations are linearised by statistical linearisation. Then, the estimation equation of the unknown input is derived based on the weighted least squares method. The multiple suboptimal fading factor is introduced into a priori error covariance matrix to improve the tracking ability for the inaccuracy of the system model and the abrupt change of state variables caused by unknown inputs. Finally, based on the unbiased minimum variance estimation, the unbiased state estimation and the error covariance matrix are derived. Singular value decomposition is performed on the error covariance matrix to improve the stability of the algorithm. Simulated results validate the effectiveness of the proposed method.

Determining the covariance matrix for a nonlinear implicit multivariate measurement equation uncertainty analysis

The application of the Guide to the Expression of Uncertainty in Measurement (GUM) for multivariate measurand equations requires an expected vector value and a corresponding covariance matrix in order to accurately calculate measurement uncertainties for models that involve correlation effects. Typically in scientific metrology applications the covariance matrix is estimated from Monte Carlo numerical simulations with the assumption of a Gaussian joint probability density function, however this procedure is often times considered too complex or cumbersome for many practicing metrologists in industrial metrology calibration laboratories, and as a result a problem which occurs is that correlation effects are frequently omitted so that uncertainties are approximated through a simple root-sum-square of uncertainties which leads to inaccuracies of measurement uncertainties. In this paper, a general purpose deterministic approach is developed using a computer algebra system (CAS) approach that avoids the need for Monte Carlo simulations in order to analytically construct the covariance matrix for arbitrary nonlinear implicit multivariate measurement models. An illustrative example for a multivariate Sakuma-Hattori pyrometer equation with the proposed method is demonstrated with explanations of underlying Python code.

A Novel Localization Method of Partial Discharge Sources in Substations Based on UHF Antenna and TSVD Regularization

Ultra-high frequency (UHF) technology has been widely used in partial discharge (PD) source localization based on the time difference of arrival (TDOA). However, the TDOA errors will then lead to the ill-conditioned problem of matrix A. A larger condition number will then lead to a more ill severe condition of linear equations <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AX</i> =b, and resulting in a decline in the localization accuracy. In this paper, a novel method for measuring the PD source has been developed based on the UHF antenna and the truncated singular value decomposition (TSVD) regularization method. First, the PD signals are obtained through the cooperation of five UHF antennas. Second, a spherical model linear transformation method is proposed for non-linear measurement equations. Then, to address the ill-conditioned problem of the measurement equations caused by the UHF antenna installation position, the linear measurement equations are improved with the pretreatment method. Finally, the TSVD regularization algorithm is applied to solve the UHF measurement equations with the ill-conditioned problems. The regularization parameter of the TSVD is then optimized by generalized cross-validation (GCV). In addition, the algorithm is verified by theoretical verification and field tests. The experimental results reveal that the proposed localization method can accurately locate the PD source.

Adaptive Transformed Unscented Simplex Cubature Kalman Filter for Target Tracking

ABSTRACT This paper peruses the Doppler-bearing passive target tracking (DBT), a task that is subject to the nonlinear measurement equations. These measurement equations form the target tracking a nontrivial duty. In order to defeat these problems, many studies have reported various nonlinear estimation techniques, such as sigma-point Kalman filters. These types of filters have demonstrated superior performances for nonlinear system parameter estimation. Consequently, the main contribution of this paper is to improve the performance of the sigma-point Kalman filters. Thereafter, this paper introduces a transformed unscented simplex cubature Kalman filter, in which adaptation techniques are utilized to reduce the dependency of the proposed filter performance on the priori statistical information of the process and measurement noises. Moreover, the proposed filter is initialized by the range-parameterised Unscented Kalman filter. The Cramér–Rao lower bound (CRLB) determines the best accessible accuracy of estimators, which is utilized in this paper to accredit the performance of the proposed filter. The validity and feasibility of the proposed filter are confirmed by Monte Carlo simulations.

Set-valued state estimation of nonlinear discrete-time systems with nonlinear invariants based on constrained zonotopes

This paper presents new methods for set-valued state estimation of discrete-time nonlinear systems whose trajectories are known to satisfy nonlinear equality constraints, called invariants (e.g., conservation laws). Set-valued estimation aims to compute tight enclosures of the possible system states in each time step subject to unknown-but-bounded uncertainties. Most existing methods employ a standard prediction-update framework with set-based prediction and update steps based on various set representations and techniques. However, achieving accurate enclosures for nonlinear systems remains a significant challenge. This paper presents new methods based on constrained zonotopes that improve the standard prediction-update framework for systems with invariants by adding a consistency step. This new step uses invariants to reduce conservatism and is enabled by new algorithms for refining constrained zonotopes based on nonlinear constraints. This paper also presents significant improvements to existing prediction and update steps for constrained zonotopes. Specifically, new update algorithms are developed that allow nonlinear measurement equations for the first time, and existing prediction methods based on conservative approximation techniques are modified to allow a more flexible choice of the approximation point, which can lead to tighter enclosures. Numerical results demonstrate that the resulting methods can provide significantly tighter enclosures than existing zonotope-based methods while maintaining comparable efficiency.

Magnetic position estimation using optimal sensor placement and nonlinear observer for smart actuators

This paper focuses on position estimation in smart actuators using non-contacting magnetic sensors. Magnetic position estimation in long-stroke actuators involves nonlinear non-monotonic measurement equations and the need to use more than one magnetic sensor. These challenges are addressed by developing a methodology for computing optimal sensor placement and an improved methodology for a LMI-based observer design technique. First, the lack of a constant gain observer with global stability for this application is shown due to the non-monotonic nature of the output nonlinearity. A switched-gain observer is therefore designed to ensure global stability. Second, the minimum singular values of the observability matrix over the range of actuator positions are utilized as a metric for optimizing sensor location and deciding on the number of sensors that need to be utilized. While a zero-norm problem formulation is non-convex and computationally expensive, the optimization problem is relaxed into a convex form and solved using a randomized rounding algorithm. Extensive experimental results are presented on the performance of the developed estimation algorithms. A nonlinear observer with two sensors placed at reasonable but non-optimal locations is shown to provide ∓ 2% error in position estimation and superior robustness properties compared to an extended Kalman filter. The use of optimal sensor locations with two sensors reduces the estimation error to less than 1% and helps meet desired performance specifications. Utilizing three sensors with optimal locations results in less than 0.5% peak position errors.

Linear Programming Contractor for Interval Distribution State Estimation Using RDM Arithmetic

State estimation (SE) of distribution networks heavily relies on pseudo measurements that introduce significant errors, since real-time measurements are insufficient. Interval SE models are regularly used, where true values of system states are supposed to be within the estimated intervals. However, conventional interval SE algorithms cannot consider the correlations of same interval variables in different terms of constraints, which results in overly conservative estimation results. In this paper, we propose a new interval SE model that is based on the relative distance measure (RDM) arithmetic. In the proposed model, measurement errors are assumed to be bounded in given sets and the state variables are described as RDM variables. Since the SE model is a non-convex, the solution's credibility cannot be guaranteed. Therefore, each nonlinear measurement equation in the model is transformed into dual inequality linear equations using the mean value theorem. The SE model is finally reformulated as a linear programming contractor that iteratively narrows the upper and lower bounds of the estimated state variables. Numerical tests on IEEE three-phase distribution networks show that the proposed method outperforms the conventional interval-constrained propagation, modified Krawczyk-operator and optimization based interval SE methods.

수정된 불변 확장칼만필터의 항법시스템에의 적용

In this paper, the application of a modified IEKF (Invariant Extended Kalman Filter) to a navigation system is studied Recently, a modification of IEKF was proposed by Seong [15] in order to consider various measurement equations, including nonlinear equations. In this paper, an aided navigation filter considering both linear and nonlinear measurement equations using the modified IEKF is presented. For nonlinear measurements such as speed and distance, the relationship between the invariant error and the measurement is derived, and from this, the linearized measurement matrix contributing to filter gain is then derived. Simulations and experiments were performed in order to verify the usefulness of the designed filter by comparison with conventional indirect feedback EKF.

Attitude dynamics aiding for three-dimensional passive target tracking of strap-down seeker based on instrumental variable Kalman filter

The accuracy of three-dimensional (3D) passive target tracking under strap-down system is usually affected by the measurement accuracy of attitude angular rate and attitude angle. In order to save the problem, a novel 3D passive target tracking method based on instrumental variable Kalman filter (IVKF) aided by the attitude dynamic is proposed. At first, the maneuvering target motion model is established based on the “current” statistical model and the filtering equation of MEMS inertial measurement unit (IMU) is also set up. Then, linearize the nonlinear state equations and replace the nonlinear measurement equations with pseudolinear equations. The 3D pseudolinear Kalman filter (PLKF) algorithm is derived according to the linear Kalman filter (KF). To counter the severe bias problems with PLKF, bias compensation and recursive instrumental variable (IV) methods are considered. In order to enhance observability of the system, a 3D motion tracking sliding-mode guidance law is deduced. Finally, some mathematical simulations were made to verify the effectiveness of the proposed method. The simulation results show the effect of the measurement accuracy and complexity of the algorithm are reduced, which proves the validity of the method.

Parameterization-switching GNSS attitude determination considering the success rate of ambiguity resolution

The two key elements of GNSS attitude determination are the correct integer ambiguity resolution (IAR) and the choice of appropriate method. The least-squares ambiguity decorrelation adjustment method is normally used to fix ambiguities using float estimates. However, it is difficult to get all ambiguities fixed in real time. Therefore, a method of partial ambiguity resolution which considers the success rate of IAR and makes full use of the ambiguity information at each epoch is proposed. The ambiguities after decorrelation can be divided into fixed ambiguities and unfixed ambiguities considering the fixing success rate. At the next epoch, the fixed ones are treated as constants, while the float estimates of the unfixed ones, along with the corresponding covariance matrix, are treated as prior information or pseudo-measurements. The parameters of the attitude determination problem can be baseline coordinates (BCs) or attitude angles (AAs). BC parameterization, without considering the baseline constraints, has the merit of resulting in a linear model; however, it involves parameter redundancy, which reduces the model strength. AA parameterization, although avoiding parameter redundancy, brings in nonlinearity and hence linearization errors in the least-squares solution. The linearization errors decrease as the number of fixed ambiguities increases. A switching-parameterization method is proposed; namely, at any epoch, as long as there are at least three fixed double-difference ambiguities, AA parameterization is adopted, otherwise BC is adopted. Even with AA, BC is firstly estimated and then transformed to attitude estimates at which the nonlinear measurement equations are linearized and solved. To verify the performance of the proposed method, a comparative study is implemented with BC and AA methods in the static experiment. The results show that the proposed method can fix ambiguity faster and have better stability and accuracy. In addition, the yaw angle is consistent with the actual running route of the vehicle in the kinematic experiment.

A Switched-Gain Nonlinear Observer for LED Optical Communication

Alignment for maintaining Line-of-Sight (LoS) between the receiver and the transmitter in a LED optical communication system is a challenging problem due to the constant movement of the underlying optical platform that is caused by vibration effects and atmospheric turbulence. In this paper, we propose a robust switched-gain nonlinear observer to estimate the angular position and velocity of the receiver orientation, which is used subsequently to follow the receiver orientation. The optical communication system model involves highly nonlinear and non-monotonic output measurement equations. Furthermore, tracking the receiver with perfect alignment results in loss of local observability. Hence, a two-receiver system is utilized to provide a robust tracking system that retains observability over the entire range of operation of the system. Lyapunov function-based analysis that ensures global stability is used to design the observer; then, a static feedback controller drives the receiver orientation based on the state estimate. Finally, numerical simulations are presented on the performance of the observer-based static feedback control in accurately estimating and tracking the angular position and velocity of the receiver orientation.