State Estimation Of Nonlinear Systems Research Articles

Adaptive Kernel Learning Kalman Filtering With Application to Model-Free Maneuvering Target Tracking

Kernel method is a non-parametric linearization method for system modeling, which uses nonlinear projection from input data space to high-dimensional Hilbert feature space and employs kernel function for hiding the projection operator in a linear learner to replace inner product calculation in Hilbert space and to avoid the curse of dimensionality. Kernel method is data-driven and learnable to nonlinear model. For independency to a priori model, it provides a novel way of state estimation for nonlinear dynamic systems with model uncertainties. In this paper, an adaptive kernel learning Kalman filtering method is proposed and applied into the problem of maneuvering target tracking. Without use of a priori system model, the method performs Kalman filtering in a reproducing kernel Hilbert space (RKHS) by estimating conditional embedding operator (CEO) as system state transfer function from training data. For the stochastic uncertainty in system model, the maximum correntropy criterion (MCC) is introduced to obtain kernel parameter optimization and balance of estimation performance, while a sliding window is designed for online updating estimation of state transfer function to get adaptability to unknown system dynamics. Such the construction for kernel Kalman filtering (KKF) is helpful to extend application from time-series signal processing to state estimation of uncertain dynamical systems. Simulation scenarios include average sunspot prediction, hovering target tracking and hypersonic maneuvering target tracking, corresponding to verification in low-dynamic, periodic-dynamic and high-dynamic systems. Numerical results have illustrated that the proposed adaptive KKF can realize model-free tracking for target that has nonlinear dynamical motion, shown adaptability to non-cooperative target maneuver and better precision and convergence speed than typical model-based algorithms.

Variational-Based Nonlinear Bayesian Filtering With Biased Observations

State estimation of dynamical systems is crucial for providing new decision-making and system automation information in different applications. However, the assumptions on the standard computational models for sensor measurements can be violated in practice due to different types of data abnormalities such as outliers and biases. In this work, we focus on the occurrence of measurement biases and propose a robust filter for their detection and mitigation during state estimation of nonlinear dynamical systems. We model the presence of bias in each dimension within the generative structure of the state-space models. Subsequently, employing the theory of Variational Bayes and general Gaussian filtering, we devise a recursive filter which we call the Bias Detecting and Mitigating (BDM) filter. As the error detection mechanism is embedded within the filter structure its dependence on any external detector is obviated. Simulations verify the performance gains of the proposed BDM filter compared to similar Kalman filtering-based approaches in terms of robustness to temporary and persistent bias presence.

An adaptive cubature Kalman filter for nonlinear systems against randomly occurring injection attacks

The problem of state estimation for nonlinear dynamic systems in the presence of randomly occurring injection attacks (ROIAs) is investigated. This paper requires no prior statistical information of the attacks, which relaxes the assumption of the existing result that the attack probability and the probability density function of attack signals need to be known. With the distribution of the attack probability and attack signals modeled as Beta distribution and Gaussian mixture distribution, a variational Bayesian based adaptive cubature Kalman filter is proposed to approximate the joint posterior distribution of the system state vector and unknown parameters. In addition, the update rules of the state and the statistical parameters of attacks are analytically derived by employing the fixed-point iteration approach. Finally, the effectiveness of the proposed filter is validated through numerical results.

Event-based state and fault estimation for stochastic nonlinear system with Markov packet dropout

This paper investigates the event-based state and fault estimation problem for stochastic nonlinear system with Markov packet dropout. By introducing the fictitious noise, the fault is augmented to the system state. Then combining the unscented Kalman filter (UKF) with event-triggered and Markov packet dropout, the modified UKF is proposed to estimate the state and fault. Meanwhile, the stochastic stability of the proposed filter is also discussed. Finally, two simulation results illustrate the performance of the proposed method.

An improved unscented Kalman filter for nonlinear systems with one-step randomly delayed measurement and unknown latency probability

In this paper, an improved unscented Kalman filter is presented to achieve state estimation for nonlinear systems with one-step randomly delayed measurement and unknown latency probability. Firstly, a Bernoulli random variable is introduced to describe the situation of randomly delayed measurement. Then, a joint prior probability density function is obtained. Finally, in the improved unscented Kalman filter algorithm, the state vector and the latency probability are estimated by utilizing a variational Bayesian technique, and the covariance matrices are computed by using the unscented transform. A univariate nonstationary growth model and a Lorenz system are included to verify that the proposed algorithm can not only implement enhanced estimation accuracy compared with the existing methods but also provide the latency probability accurately.

MultiPDF particle filtering in state estimation of nonlinear objects

This paper presents a new particle filter algorithm (MultiPDF) for state estimation of nonlinear systems. The proposed method is a modification of the standard particle filter approach. Due to the strong need for the acceleration of calculations and an improvement in the estimation quality of state estimation, the authors propose a method which enables one to divide the main particle filter into smaller sub-filters with an accordingly smaller number of particles for each one of them. The algorithm has been implemented for various numbers of particles and subordinate parallel filters. Estimation quality has been checked for nine nonlinear objects (both one- and multidimensional) and evaluated through the quality index, average root-mean-squared error. The computation time of the particle filter algorithm for several hardware configurations has been compared. Based on the obtained results, it can be concluded that, besides the computation acceleration, the parallelization of the particle filter’s operation also improves the estimation quality.

Optimal path tracking control for intelligent four-wheel steering vehicles based on MPC and state estimation

Four-wheel steering (4WS) vehicles have better stability control and path following performance than front-wheel steering (FWS) vehicles. Aiming at this characteristic, a new four-wheel active steering control strategy is proposed. For intelligent vehicle path tracking and nonlinear vehicle system state estimation, a path tracking control algorithm based on the MPC algorithm is designed to analyze the stability of the vehicle and set up constraints to achieve accurate tracking of the reference path. Automotive dynamic control systems require information on system variables. For example, the sideslip angle of electric vehicles cannot be measured directly. Based on UKF theory and the information input of low-cost sensors on the vehicle, an estimator is designed to estimate vehicle sideslip angle and yaw rate. According to the difference between the estimated value and the ideal value of the vehicle state, the LQR optimal controller is designed to realize the optimal control of the front and rear steering. And compared with the dynamic simulation results of front-wheel steering, proportional control four-wheel steering and yaw rate feedback four-wheel steering. The experimental results of the simulation platform show that the path tracking 4WS state feedback optimal control method has good lateral control stability and path tracking accuracy.

Design of the modified fractional central difference Kalman filters under stochastic colored noises

For state estimation of discrete nonlinear fractional stochastic systems, this study presents two innovative modified fractional central difference Kalman filters. We consider a complicated scenario where the process noise or measurement noise in the system become colored noise. Firstly, the nonlinear function is linearized by utilizing the Stirling polynomial interpolation formula. Thus there is no need to calculate the Jacobi matrix for both algorithms, which means very few application limitations. Then, based on the augmented-state method, we develop an augmented state fractional central difference Kalman filter under the scenario of colored process noise. Afterwards, a state estimation algorithm for handling stochastic systems containing colored measurement noise is put forward by using the measurement expansion method. Finally, to perform the superiority of the developed algorithms, several simulations are carried out. As well, the algorithms derived in this paper are contrasted with the original fractional central difference Kalman filter and three other algorithms. Notably, a simulation with engineering significance for the state-of-charge estimation for lithium-ion batteries is also introduced, Aside from the commonly used numerical simulation. The results verify the superiority of the developed algorithms in sense of estimation accuracy and real-time performance.

Distributed trust‐based unscented Kalman filter for non‐linear state estimation under cyber‐attacks: The application of manoeuvring target tracking over wireless sensor networks

This paper is concerned with secure state estimation of non-linear systems under malicious cyber-attacks. The application of target tracking over a wireless sensor network is investigated. The existence of rotational manoeuvre in the target movement introduces non-linear behaviour in the dynamic model of the system. Moreover, in wireless sensor networks under cyber-attacks, erroneous information is spread in the whole network by imperilling some nodes and consequently their neighbours. Thus, they can deteriorate the performance of tracking. Despite the development of target tracking techniques in wireless sensor networks, the problem of rotational manoeuvring target tracking under cyber-attacks is still challenging. To deal with the model non-linearity due to target rotational manoeuvres, an unscented Kalman filter is employed to estimate the target state variables consisting of the position and velocity. A diffusion-based distributed unscented Kalman filtering combined with a trust-based scheme is applied to ensure robustness against the cyber-attacks in manoeuvring target tracking applications over a wireless sensor network with secured nodes. Simulation results demonstrate the effectiveness of the proposed strategy in terms of tracking accuracy, while random attacks, false data injection attacks, and replay attacks are considered.

Study on initial value problem for fractional-order cubature Kalman filters of nonlinear continuous-time fractional-order systems

To realize the state estimation of nonlinear continuous-time fractional-order systems, two types of fractional-order cubature Kalman filters are designed to solve problem on the initial value influence. For the first type of cubature Kalman filter (CKF), the initial value of the estimated system is also regarded as the augmented state, and the augmented state equation is constructed to obtain the CKF based on Grünwald–Letnikov difference. For the second type of CKF, the fractional-order hybrid extended-cubature Kalman filter is proposed to weaken the influence of initial value by the first-order Taylor expansion and the third-order spherical-radial rule. These two methods can reduce the influence of initial value on the state estimation effectively. Finally, the effectiveness of the proposed CKFs is verified by two simulation examples.

RNN-Based Learning of Nonlinear Dynamic System Using Wireless IIoT Networks

We consider the recurrent neural network (RNN)-based remote state estimation for nonlinear dynamic systems with unknown state dynamics. The nonlinear dynamic plant is monitored by multiple distributed IIoT sensors over a random access wireless network with shared common spectrum. We focus on the remote state estimation algorithm design so as to achieve remote state estimation stability subject to noninvertible nonlinear sensor state observations, imperfect channel state information (CSI) at the remote estimator, and various wireless impairments, such as multisensor interference, wireless fading, and additive channel noise. Utilizing a state diffeomorphism, the original system is transformed into a canonical form with a linear rank deficient observation matrix. We propose a novel RNN remote state estimator based on the pole placement design associated with the transformed rank deficient state measurement matrices. We further propose a novel online training algorithm such that the RNN at the remote estimator can not only address the divergence issue over wireless networks but also effectively learn the unknown nonlinear plant dynamics despite rank deficiency and imperfect CSI. Using the Lyapunov drift analysis approach, we establish closed-form sufficient requirements on the communication resources needed to achieve almost sure stability of both state estimation and RNN online training in the high signal-to-noise ratio (SNR) regime. As a result, our proposed scheme is asymptomatic optimal for large SNR in the sense that both the plant state and the unknown plant nonlinearity can be perfectly recovered at the remote estimator. The proposed scheme is also compared with various baselines and we show that significant performance gains can be achieved.

A Novel Robust Nonlinear Kalman Filter Based on Multivariate Laplace Distribution

In this brief, we investigate the robust state estimation of nonlinear systems with heavy-tailed noise. Based on the fact that the multivariate Laplace (ML) distribution has heavier tails than Gaussian distribution but is only determined by its mean and covariance, we utilize the ML distribution to model the heavy-tailed measurement noise. Based on the Gaussian mixture form of ML distribution, the system state and the covariance of ML distribution are inferred by variational Bayesian method, and then a closed recursive robust nonlinear Kalman filter is obtained with the nonlinear integrations calculated by numerical integration methods. Simulation results demonstrate that the proposed algorithm is insensitive to its free parameters and can obtain better estimation accuracy than related robust algorithms.

A derivative-free nonlinear Kalman filtering approach using flat inputs

ABSTRACT Differential flatness theory has been widely used for both tracking control and state estimation of nonlinear systems. Recently, the concept of flat inputs has been proved to be a valuable tool for control design if the system is non-differentially flat. However, flat inputs have not been used yet to solve state estimation problems for these systems. By constructing flat inputs, we introduced a novel state estimation-based control strategy for both observable non-differentially flat nonlinear systems and differentially flat nonlinear systems whose flat output vector is not a measurable variable, as long as the internal dynamics of the system are stable. The efficiency of the proposed approach is illustrated through two numerical examples. Our findings indicated that we increased the class of systems to which the derivative-free nonlinear Kalman filtering based on differential flatness theory can be applied.

Distributed resilient state estimation for nonlinear systems with randomly occurring communication delays and missing measurements

SummaryThis article is concerned with the distributed H∞ resilient state estimation problem for a class of nonlinear systems with randomly occurring communication delays and missing measurements in sensor networks. A novel sensor model is proposed, in which two Bernoulli distributed white sequences are introduced to describe the random communication delay and missing measurements in a unified framework. Meanwhile, the estimator gain is allowed to fluctuate within a certain range. Based on the developed model, a novel Lyapunov–Krasovskii functional with multiple delay information terms is constructed, then the stochastic analysis technique and the extended integral inequality are used to calculate the functional derivative. Consequently, the existence conditions for the required distributed estimator are established to ensure that the estimation error system is asymptotically mean‐square stable and satisfies the prescribed H∞ performance constraint, and the desired gain of distributed resilient estimator is also solved by linearizing the nonlinear terms. Finally, a numerical example is given to illustrate the effectiveness of the proposed algorithm.

Set-valued mode recognition-based Bayesian estimation for nonlinear stochastic systems with unknown sensor mode

This paper proposes the problem of joint state estimation and mode recognition for nonlinear stochastic systems with unknown sensor mode. The considered sensor mode is represented by a random finite set, whose elements can be one specific mode or a set of certain modes. A set-valued mode recognition-based Bayesian estimation framework is proposed to propagate the posterior density of the state conditioned on sensor modes and measurements, where the mode is recognized based on the maximum correntropy criterion. Furthermore, a mode-separability metric is proposed to discern the reliability of mode recognition, and utilized to derive two distinct implementation schemes, including state estimation based on separable and inseparable modes. Simulation results of fault detection and target tracking are provided to demonstrate the superiority of the proposed method in terms of state estimation accuracy and mode recognition effectiveness.

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.

Recursive joint Cramér‐Rao lower bound for parametric systems with two‐adjacent‐states dependent measurements

Joint Cramér-Rao lower bound (JCRLB) is very useful for the performance evaluation of joint state and parameter estimation (JSPE) of non-linear systems, in which the current measurement only depends on the current state. However, in reality, the non-linear systems with two-adjacent-states dependent (TASD) measurements, that is, the current measurement is dependent on the current state as well as the most recent previous state, are also common. First, the recursive JCRLB for the general form of such non-linear systems with unknown deterministic parameters is developed. Its relationships with the posterior CRLB for systems with TASD measurements and the hybrid CRLB for regular parametric systems are also provided. Then, the recursive JCRLBs for two special forms of parametric systems with TASD measurements, in which the measurement noises are autocorrelated or cross-correlated with the process noises at one time step apart, are presented, respectively. Illustrative examples in radar target tracking show the effectiveness of the JCRLB for the performance evaluation of parametric TASD systems.

State Estimation and Stabilization of Nonlinear Systems with Sampled-Data Control and Uncertain Disturbances

State Estimation and Stabilization of Nonlinear Systems with Sampled-Data Control and Uncertain Disturbances

Constrained State Estimation for Nonlinear Systems: A Redesign Approach Based on Convexity

Given a plant whose trajectories of interest remain in a known compact set and an associated observer, we propose a general framework to modify this observer so that its state remains in a given convex set for all times, without altering the observer guaranteed performances in terms of convergence and robustness to external disturbances. The methodology can be applied to any time-varying continuous-time, discrete-time, and hybrid system/observer, for which a quadratic Lyapunov function is used for the analysis. The proposed approach is relevant, for instance, to remove the peaking phenomenon, to attenuate the effect of impulsive outliers in the measurement, to avoid aberrant estimates during transients, or to guarantee a given range for variables in embedded systems.

Joint Sensor Node Selection and State Estimation for Nonlinear Networks and Systems

State estimation and sensor selection problems for nonlinear networks and systems are ubiquitous problems that are important for the control, monitoring, analysis, and prediction of a large number of engineered and physical systems. Sensor selection problems are extensively studied for linear networks. However, less attention has been dedicated to networks with nonlinear dynamics. Furthermore, widely used sensor selection methods relying on structural (graph-based) observability approaches might produce far from optimal results when applied to nonlinear network dynamics. In addition, state estimation and sensor selection problems are often treated separately, and this might decrease the overall estimation performance. To address these challenges, we develop a novel methodology for selecting sensor nodes for networks with nonlinear dynamics. Our main idea is to incorporate the sensor selection problem into an initial state estimation problem. The resulting mixed-integer nonlinear optimization problem is approximately solved using three methods. The good numerical performance of our approach is demonstrated by testing the algorithms on prototypical Duffing oscillator, associative memory, and chemical reaction networks. The developed codes are available online.