An improved unscented Kalman filter for nonlinear systems with one-step randomly delayed measurement and unknown latency probability
Introduction
State estimation has received wide attention because of the application in various real-time control systems. Reliable estimation of states contributes to provide the accurate and safe control of systems [1], [2], [3], [4]. As a minimum variance estimator, the Kalman filter (KF) is a popular and efficient algorithm for linear dynamic systems with Gaussian noise to achieve the real-time online state estimation [5], [6], [7], [8]. The Kalman filter and its improved filters have been proposed to address various filtering problems of linear systems [9], [10], [11], [12], [13]. However, nonlinear system is more common in practice. The extended Kalman filter (EKF) was proposed to achieve estimation for nonlinear systems. In the EKF algorithm, the nonlinear state-space equation is linearized by using the Taylor formula, and local linearization is used to approximate the non-linearities of the system dynamics [14], [15]. The unscented Kalman filter (UKF) is proposed to statistically process sigma sampling points and propagate mean and covariance information of a nonlinear system by utilizing the unscented transform (UT) [16], [17], [18]. Nevertheless, it is inevitable for the existence of randomly delayed measurements (RDMs) while signals and data are transmitted through the channel in many spacecrafts control [19], target tracking [20], and communication [21] systems, and the above estimators suffer from performance deterioration in these scenarios.
To solve RDMs of nonlinear systems, some methods were proposed. A novel Gaussian filter was proposed based on Gaussian approximation (GA), which provided a general estimation framework applied to both linear and nonlinear systems [22]. In [23], an improved Bayesian filtering approach was proposed, in which the likelihood function was computed by marginalizing out the delay variable to extract accurate information from the delayed measurements. In [24], a novel Gaussian smoother was proposed based on the general GA and the Gaussian mixture approximation, in which the UT was utilized to address nonlinearity problem. In [25], a new particle smoother was proposed, in which the samples were randomly drawn to calculate the delayed posterior density, and the weights of major particles were updated based on Bayesian approach. Above methods are implemented based on the known delay probability. However, the delay probability is unknown and time-varying in practice due to network congestion in the process of information transmission [27]. Therefore, the estimation accuracy of these methods may be reduced by using an inaccurate delay probability.
To solve RDMs and unknown delay probability of nonlinear systems, a novel Rauch-Tung-Striebel smoother was proposed, in which the delay probability was estimated based on the expectation maximization (EM) framework [26]. To improve the performance of the method further, the above smoother with a moving window was proposed in [27]. However, the two smoothers are not utilized to achieve estimation. In [28], a novel particle filter was proposed based on maximizing the likelihood criterion. A novel UKF was proposed based on the Bayesian filtering approach, in which the sigma points were obtained by utilizing proportional symmetric sampling strategy [29]. Since the latency probability is assumed to be constant in these methods, the time-varying unknown delayed measurements probability is still not considered.
Recently, an improved KF was proposed to achieve the adaptive estimation of delay probability based on the variational Bayesian (VB) [30]. It demonstrates good performance in the linear systems with RDMs and changing unknown delay probability. However, it cannot be directly applied into the nonlinear systems. Motivated by the method, we focus on a nonlinear system with RDMs and unknown delay probability, an improved UKF is proposed to solve the online filtering problem for nonlinear systems. In the improved UKF algorithm, an augmented state is defined and corresponding distribution model is deduced. Then, the joint prior probability density function (PDF) is computed. The posterior PDF of the augmented state vector is updated by employing the state augmentation approach. The state vector and the time-varying latency probability are estimated by employing the VB. The covariance matrices are computed by utilizing the UT. Through the simulations of the univariate nonstationary growth model (UNGM) and Lorenz system, it is obvious that the improved UKF can not only achieve the enhanced estimation accuracy compared with the UKF but also provide the latency probability accurately.
The remainder of this paper is organized as follows. In section 2, the problem definition is given. In section 3, improved UKF is designed. Simulations are performed to verify the effectiveness of the proposed method in section 4. In section 5, conclusions are drawn.
Section snippets
Problem formulation
Consider a discrete-time state space model (SSM) of the nonlinear system with RDM which is described by [27] where k refers to the discrete time index, and are two known nonlinear functions, denotes the state vector, denotes the ideal measurement vector, is the process noise vector and , is the measurement noise vector and , and are uncorrelated zero-mean Gaussian white
Main results
In this section, the improved UKF is implemented by utilizing the VB to estimate the state and latency probability for system (1)–(3).
Simulations
In this section, to verify the effectiveness of the proposed method, the improved UKF is compared with the UKF through some comparative experiments. Simulations are performed in the univariate nonstationary growth model and the Lorenz system.
Conclusion
In this paper, an improved UKF was proposed for nonlinear systems with one-step RDM and unknown latency probability. A joint prior PDF about the augmented state vector, the latency probability and the Bernoulli variable is obtained. The latency probability and the augmented state vector were estimated utilizing the VB and state augmentation approach, and the covariance matrices were computed by using the unscented transform. The proposed method was applied to the univariate nonstationary growth
CRediT authorship contribution statement
Yuze Tong: Conceptualization, Formal analysis, Methodology, Software. Zongsheng Zheng: Data curation, Validation, Writing – original draft. Wenli Fan: Writing – review & editing. Quanyou Li: Investigation. Zhigang Liu: Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported in part by the General Project of Humanities and Social Sciences Research of the Ministry of Education under Grant 18YJCZH028 and 20XJCZH004 and the Fundamental Research Funds for the Central Universities under Grant A0920502052001-1, and Chengdu soft science research project under Grant 2020-RK00-00367-ZF.
Yuze Tong received the B.S. degree from Lanzhou Jiaotong University, Lanzhou. China, in 2016-2020. Currently, she is working towards the M.S. degree at Southwest Jiaotong University, China. Her research focuses on parameter and state estimation.
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Yuze Tong received the B.S. degree from Lanzhou Jiaotong University, Lanzhou. China, in 2016-2020. Currently, she is working towards the M.S. degree at Southwest Jiaotong University, China. Her research focuses on parameter and state estimation.
Zongsheng Zheng received the B.S. degree in bioinformatics and the Ph.D. degree in electrical engineering from Southwest Jiaotong University, Chengdu, China, in 2013 and 2020, respectively. During 2018-2019, he was a Visiting Scholar at the Bradley Department of Electrical and Computer Engineering at Virginia Tech-Northern Virginia Center, Falls Church, VA, USA. He is currently an Associate Research Fellow at the College of Electrical Engineering, Sichuan University. His research interests include uncertainty quantification, parameter and state estimation.
Wenli Fan received the Ph.D. degree in Power System and its Automation from Southwest Jiaotong University of China in 2014.
He is currently a Lecturer with the School of Electrical Engineering, Southwest Jiaotong University. His current research interests include power system analysis and control, especially power system complexity and security.
Quanyou Li received the B.S. degree from Yangtze University, China, in 2020. Currently, he is working towards the M.S. degree at Southwest Jiaotong University, China. His current research focuses on state assessment of power system.
Zhigang Liu (Senior Member, IEEE) received the Ph.D. degree in power system and automation from Southwest Jiaotong University, Chengdu, China, in 2003. He is currently a Full Professor with the School of Electrical Engineering, Southwest Jiaotong University. His current research interests include electrical relationship of vehicle grid in high-speed railways, power quality considering grid connect of new energies, pantograph-catenary dynamics, fault detection, status assessment, and active control. Dr. Liu was selected as a fellow of The Institution of Engineering and Technology (IET) in 2017. He is also an Associate Editor of the IEEE Transactions on Instrumentation and Measurement, the IEEE Transactions on Vehicular Technology, and IEEE Access.