State Estimation Problem Research Articles

Model-free distributed state estimation with local measurements.

In this paper, the state estimation problem of physical plants with unknown system dynamic is revisited from the perspective of limited output information measurement, which corresponds to those with characteristics of high-dimensional, wide-area coverage and scatter. Given this fact, a network of sensors are used to carry out the measurement with each one accessing only partial outputs of the targeted systems and a novel model-free state estimation approach, named distributed stochastic variational inference state estimation, is proposed. The key idea of this method is to compensate for the impacts of local output measurements by adding nearest-neighbor rule-based information interaction among estimators to complete the state estimation. It finds from the numerical experiments that the proposed method has clear advantages in both estimation accuracy and speed, and it also provides guidance on how to improve the efficiency of state estimation under local measurements.

A Closed-Form Solution to Observer Design Problem for Ostensible Metzler Takagi-Sugeno Systems

This paper addresses the state estimation problems related to the generalized fuzzy observer design for ostensible Metzler Takagi-Sugeno (T-S) systems. Attention is focused on design constraints for the concept of diagonal stabilization and positivity of observer gain matrices. On the basis of some new interpretations, the parameterizations of ostensible Metzler T-S fuzzy systems is presented, which opens the way to the solution of the design problem using only the principle of linear matrix inequalities. The same approach is intended to ensure the stability of the dynamics of the estimation error. The presented method extends and generalizes the results that have been presented in the literature so far.

Consensus-Based Power System State Estimation Algorithm Under Collaborative Attack.

Due to its vulnerability to a variety of cyber attacks, research on cyber security for power systems has become especially crucial. In order to maintain the safe and stable operation of power systems, it is worthwhile to gain insight into the complex characteristics and behaviors of cyber attacks from the attacker's perspective. The consensus-based distributed state estimation problem is investigated for power systems subject to collaborative attacks. In order to describe such attack behaviors, the denial of service (DoS) attack model for hybrid remote terminal unit (RTU) and phasor measurement unit (PMU) measurements, and the false data injection (FDI) attack model for neighboring estimation information, are constructed. By integrating these two types of attack models, a different consensus-based distributed estimator is designed to accurately estimate the state of the power system under collaborative attacks. Then, through Lyapunov stability analysis theory, a sufficient condition is provided to ensure that the proposed distributed estimator is stable, and a suitable consensus gain matrix is devised. Finally, to confirm the viability and efficacy of the suggested algorithm, a simulation experiment on an IEEE benchmark 14-bus power system is carried out.

Motion State Estimation with Bandwidth Constraints and Mixed Cyber-Attacks for Unmanned Surface Vehicles: A Resilient Set-Membership Filtering Framework.

This paper investigates the motion state estimation problem of the unmanned surface vehicle (USV) steering system in wireless sensor networks based on the binary coding scheme (BCS). In response to the presence of bandwidth constraints and mixed cyber-attacks in USV communication networks, this paper proposes an improved set-membership state estimation algorithm based on BCS. This algorithm partially addresses the problem of degraded performance in USV steering motion state estimation caused by mixed cyber-attacks and bandwidth constraints. Furthermore, this paper proposes a robust resilient filtering framework considering the possible occurrence of unknown but bounded (UBB) noises, model parameter uncertainties, and estimator gain perturbations in practical scenarios. The proposed framework can accurately estimate the sway velocity, yaw velocity, and roll velocity of the USV under the concurrent presence situation of mixed cyber-attacks, communication capacity constraints, UBB noises, model parameter uncertainties, and estimator gain perturbations. This paper first utilizes mathematical induction to provide the sufficient conditions for the existence of the desired estimator, and obtains the estimator gain by solving a set of linear matrix inequalities. Then, a recursive optimization algorithm is utilized to achieve optimal estimation performance. Finally, the effectiveness of the proposed estimation algorithm is verified through a simulation experiment.

Stubborn State and Disturbance Observer Co‐Design for Nonlinear Descriptor Systems With δQC$$ \delta \mathrm{QC} $$ via a Dynamic Event‐Triggered Mechanism

ABSTRACTThis article studies the state and disturbance simultaneous estimation problem for a class of nonlinear descriptor systems with using a dynamic event‐triggered mechanism. refers to incremental quadratic constraints, which can provide a unified description of many types of common nonlinear functions. To reduce the negative impact of measurement outliers on the identification estimation, a stubborn state and disturbance observer co‐design scheme is proposed for the first time by embedding dynamic saturation output estimation errors. Meanwhile, a dynamic event‐triggered mechanism is introduced to avoid the need for continuously available output information, which can reduce the pressure on communication resources. By constructing a Lyapunov function, existence conditions of the stubborn state and disturbance observer are obtained in the form of a convex optimization problem so that the error estimation maintains an acceptable estimation performance. Finally, simulation examples illustrate the universality and stubbornness of the proposed observer.

The Square-Root Unscented and the Square-Root Cubature Kalman Filters on Manifolds.

Estimating the state of a system by fusing sensor data is a major prerequisite in many applications. When the state is time-variant, derivatives of the Kalman filter are a popular choice for solving that task. Two variants are the square-root unscented Kalman filter (SRUKF) and the square-root cubature Kalman filter (SCKF). In contrast to the unscented Kalman filter (UKF) and the cubature Kalman filter (CKF), they do not operate on the covariance matrix but on its square root. In this work, we modify the SRUKF and the SCKF for use on manifolds. This is particularly relevant for many state estimation problems when, for example, an orientation is part of a state or a measurement. In contrast to other approaches, our solution is both generic and mathematically coherent. It has the same theoretical complexity as the UKF and CKF on manifolds, but we show that the practical implementation can be faster. Furthermore, it gains the improved numerical properties of the classical SRUKF and SCKF. We compare the SRUKF and the SCKF on manifolds to the UKF and the CKF on manifolds, using the example of odometry estimation for an autonomous car. It is demonstrated that all algorithms have the same localization performance, but our SRUKF and SCKF have lower computational demands.

Distributed adaptive moving horizon estimation for multi-sensor networks subject to quantization effects

This paper investigates the distributed state estimation problem for multi-sensor networks with quantized measurements. Within the Bayesian framework, a distributed adaptive moving horizon estimation algorithm is developed. Unlike the existing methods regarding quantized errors roughly as bounded uncertainties, the posterior distributions of the errors are demanded to be derived. To overcome the difficulty of evaluating the posterior distributions for series of the states and quantized errors jointly, the variational Bayesian methodology is adopted to approximate the true distributions. Based on the fixed-point iteration method, the update rules are analytically derived, with the convergence criterion provided. Furthermore, by incorporating the average consensus algorithm into the prediction process, all sensors can achieve consensus on their estimates in a distributed manner. Finally, a numerical example of target tracking under logarithmic and uniform quantization effects is given to illustrate the validity of the proposed algorithm.

Distributed Consensus Filtering Over Sensor Networks With Asynchronous Measurements

ABSTRACTIn practical sensor networks, sensor nodes may operate with different sampling periods and initial sampling time instants, and their observations may also be nonuniform. Unfortunately, the research on distributed state estimation problems over such asynchronous sensor networks is very limited. Thus, this article focuses on the distributed consensus filtering problem over sensor networks with asynchronous measurements. First, the asynchronous measurement from each sensor is synchronized by the continuous‐time state evolution equation to a unified filtering fusion time instant within a given filtering period. After measurement synchronization, the statistical characteristics of measurement noise change. The cross‐correlations between the converted measurement noises are analyzed, as well as one‐step correlations between the converted measurement noises and the process noise. Second, an optimal asynchronous distributed consensus filter is designed based on synchronization measurements under the criterion of minimum mean‐square error with the above correlations between various types of noises taken into account. Meanwhile, a suboptimal distributed consensus filtering algorithm is further proposed to reduce computational complexity. Finally, based on the Lyapunov function method, the stability of the estimation error is theoretically demonstrated with an appropriate selection of consensus filtering gain and validated through simulations.

State Estimation for Measurement-Saturated Memristive Neural Networks with Missing Measurements and Mixed Time Delays Subject to Cyber-Attacks: A Non-Fragile Set-Membership Filtering Framework

This paper is concerned with the state estimation problem based on non-fragile set-membership filtering for a class of measurement-saturated memristive neural networks (MNNs) with unknown but bounded (UBB) noises, mixed time delays and missing measurements (MMs), subject to cyber-attacks under the framework of weighted try-once-discard protocol (WTOD protocol). Considering bandwidth-limited open networks, this paper proposes an improved set-membership filtering based on WTOD protocol to partially solve the problem that multiple sensor-related problems and multiple network-induced phenomena influence the state estimation performance of MNNs. Moreover, this paper also discusses the gain perturbations of the estimator and proposes an improved non-fragile estimation framework based on set-membership filtering, which enhances the robustness of the estimation approach. The proposed estimation framework can effectively estimate the state of MNNs with UBB noises, estimator gain perturbations, mixed time-delays, cyber-attacks, measurement saturations and MMs. This paper first utilizes mathematical induction to provide the sufficient conditions for the existence of the desired estimator, and obtains the estimator gain by solving a set of linear matrix inequalities. Then, a recursive optimization algorithm is utilized to achieve optimal estimation performance. The effectiveness of the theoretical results is verified by comparative numerical simulation examples.

Polynomial Fuzzy Observer-Based Feedback Control for Nonlinear Hyperbolic PDEs Systems.

This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler's homogeneous theorem, Lyapunov-Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.

Set‐membership state estimation for delayed switched systems with interval uncertainty: A zonotopic approach

SummaryConsidering the existence of time delay and interval uncertainty, a set‐membership state estimation problem based on zonotopes is studied for discrete‐time switched systems subject to unknown but bounded noises. To conform the actual situation, a state observer asynchronous with the subsystem is designed. The performance is introduced to attenuate the effect of noises in the observer design. By dealing with the interval uncertainty, the design approach of observer is given and transformed into a convex optimization problem which can be solved off‐line. The zonotope and the estimation interval containing state are provided based on the observer. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm.

Robust fixed‐point Kalman smoother for bilinear state‐space systems with non‐Gaussian noise and parametric uncertainties

SummaryKalman smoother is an effective algorithm to estimate the state of the dynamic systems with Gaussian noise. However, when the system is affected by non‐Gaussian noise, the traditional Kalman smoother may suffer severe performance degradation, since it is derived from the minimum mean square error criterion. By introducing the maximum correntropy criterion, which accounts for all higher order moments and has the ability to resist non‐Gaussian noise, this article studies the state estimation problem of the bilinear state‐space system with non‐Gaussian noises and parametric uncertainties. The bilinear system with parametric uncertainties is transformed into a linear time‐varying system, and a robust fixed‐point Kalman filter algorithm is derived based on the Cauchy kernel‐based correntropy criterion. To improve the state estimation accuracy, a Cauchy kernel‐based fixed‐point Kalman smoother (CK‐FPKS) algorithm is presented by introducing the backward smoothing. Simulation results show the effectiveness of the proposed algorithm.

State estimation of singularly perturbed Semi-Markov jump coupled neural networks: A two-time-scale event-triggered approach

The state estimation problem of singularly perturbed semi-Markov jump coupled neural networks is investigated in this paper, in which a two-time-scale event-triggered mechanism is proposed to adapt the two-time-scale character of the singularly perturbed coupled neural networks such that mitigate the burden of data communication. A dual-rate sampling strategy is used for slow and fast states such that the proposed two-time-scale event-triggered mechanism contains two independent event-triggered conditions corresponding to the slow and fast states respectively. Based on the two-time-scale event-triggered mechanism, sufficient conditions for the H∞ performance analysis of singularly perturbed semi-Markov jump coupled neural networks are established. Furthermore, under the circumstances that the singularly perturbed parameter does not exceed the presupposed upper bound, a design algorithm is developed for the two-time-scale state estimator to ensure that the error dynamics are stochastically stable. Finally, an example is used to illustrate the effectiveness of this method used in singularly perturbed semi-Markov jump coupled neural networks.

A variational Bayesian based robust filter for unknown measurement bias and inaccurate noise statistics

In many practical fields, the unknown time-varying measurement biases (additive and multiplicative bias) and heavy-tailed measurement noise caused by some unpredictable anomalous behaviors may degrade the performance of conventional Kalman filter seriously. To solve the state estimation problem of systems with time-varying measurement biases and heavy-tailed measurement noise, this paper proposes a new variational Bayesian (VB) based robust filter. Firstly, the non-Gaussian measurement likelihood probability density function (ML-PDF) with multiplicative and additive measurement bias is built. Then, the conjugate prior distributions for unknown bias and noise scale parameters are selected, and the VB method is utilized to jointly infer the system state, unknown measurement biases and inaccurate measurement noise covariance matrix. Finally, a VB based robust filter is derived and its effectiveness is verified by the numerical simulations.

Joint detection and state estimation based on GPS spoofing attack in smart grids

Phasor Measurement Units (PMUs) are widely used in power system state estimation due to its high sampling rate and real-time estimation of power state in power systems. However, it is also vulnerable to cyberattacks (GPS spoofing attacks) since it uses unencrypted GPS civil signals for synchronization. The GPS spoofing attacks (GSAs) will intentionally manipulate the time reference of PMUs, which is equivalent to falsifying the phase angle of the PMU measurements. In this paper, we consider the secure power state estimation and attacks detection with the unknown GSAs in the power systems. Particularly, we derived the PMU-based GSAs model in the power state estimation problem, which is formulated as a mixed maximum likelihood problem. In the formulated problem, the involved unknown parameters are coupled and the objective function is non-convex, which are of significant challenges in finding optimal solutions. To tackle these challenges, a joint maximum a posteriori and maximum likelihood (JMAP-ML) algorithm is proposed to securely estimate the power state and detect the GSAs in the mixed measurements of PMUs. Different testing scenarios in IEEE 14 bus system are simulated to show the proposed algorithm’ performance on GSAs detection and state estimation. Numerical examples demonstrate the improved accuracy of our algorithm compared with classical algorithms when GSAs are present. And we conclude that when a sufficient number of PMUs are deployed in the system, the impact of GSAs will be largely compensated in the estimation stage.

Convergence analysis of state estimation for UAV formation under GPS‐denied environments

AbstractIn this article, we focus on the state estimation and formation control problem for a multi‐UAV system, where only parts of UAVs have GPS measurements, that is, the anchors, and the other UAVs measure their relative positions with anchors. With a second‐order discrete‐time model of each UAV in the formation, we analyze the convergence of state estimation (including its expectation and covariance) for commonly used consensus‐based formation control protocols, under the effect of control and measurement noise. More specifically, based on the discrete‐time stability analysis, we obtain the sufficient and necessary conditions for the expectation stability of the formation control. Using the fixed point theorem in positive symmetric matrix subspace and observability analysis, we rigorously prove that the covariance of estimation is bounded if and only if the anchor node set is globally reachable. Based on our analysis, the formation control parameters and system topology can be designed to satisfy sufficient conditions, such that the desired formation pattern can be achieved for the multi‐UAV dynamic system under GPS‐denied environments. Simulation results corroborate the effectiveness of our theoretical results.

Set-membership state estimation for time-varying complex networks: two zonotopic design methods

This article studies the zonotopic set-membership state estimation problem for linear time-varying complex networks with unknown-but-bounded (UBB) noises, where the UBB noises are contained by a set of zonotopes. The objective of the addressed problem is to give two design methods, namely the correction matrix method and the state observer method, where a time-varying zonotopic sequence containing all possible states of the system is obtained. The expressions of the correction matrix and the observer gain are given under the F-radius criterion, and the desired minimum zonotopes are obtained. In addition, a state observer based on the measured output at the current moment is designed to analyze the equivalence between the above two methods. Finally, in order to demonstrate the effectiveness of the proposed state estimation algorithms, two simulation examples are presented.

CGNSDE: Conditional Gaussian neural stochastic differential equation for modeling complex systems and data assimilation

A new knowledge-based and machine learning hybrid modeling approach, called conditional Gaussian neural stochastic differential equation (CGNSDE), is developed to facilitate modeling complex dynamical systems and implementing analytic formulae of the associated data assimilation (DA). In contrast to the standard neural network predictive models, the CGNSDE is designed to effectively tackle both forward prediction tasks and inverse state estimation problems. The CGNSDE starts by exploiting a systematic causal inference via information theory to build a simple knowledge-based nonlinear model that nevertheless captures as much explainable physics as possible. Then, neural networks are supplemented to the knowledge-based model in a specific way, which not only characterizes the remaining features that are challenging to model with simple forms but also advances the use of analytic formulae to efficiently compute the nonlinear DA solution. These analytic formulae are used as an additional computationally affordable loss to train the neural networks that directly improve the DA accuracy. This DA loss function promotes the CGNSDE to capture the interactions between state variables and thus advances its modeling skills. With the DA loss, the CGNSDE is more capable of estimating extreme events and quantifying the associated uncertainty. Furthermore, crucial physical properties in many complex systems, such as the translate-invariant local dependence of state variables, can significantly simplify the neural network structures and facilitate the CGNSDE to be applied to high-dimensional systems. Numerical experiments based on chaotic systems with intermittency and strong non-Gaussian features indicate that the CGNSDE outperforms knowledge-based regression models, and the DA loss further enhances the modeling skills of the CGNSDE.

A scale-similarity based workflow for self-supervised reconstruction of small-scales in turbulence using deep learning

Deep learning has been extensively utilized for modeling and analysis of fluid turbulence. One such application is the use of super-resolution (SR) algorithms to reconstruct small-scale structures from their large-scale counterparts for turbulent flows. To date, all SR algorithms have been supervised or require unpaired reference data at a high resolution for training. This renders the model inapplicable to practical fluid flow scenarios, in which the generation of a high-resolution ground truth by resolving all scales down to the Kolmogorov scale becomes prohibitive. Hence, it is imperative to develop physics-guided models that exploit the multiscale nature of turbulence. Considering SR as a state-estimation problem, we present a self-supervised workflow based on deep neural networks to reconstruct small-scale structures that are relevant to homogeneous isotropic turbulence. In addition to visual similarity, we assessed the quality of the obtained reconstruction using spectra, structure functions, and probability density functions of the gradients of velocity and a passive scalar. From the analysis, we infer that the outputs of the workflow are in statistical agreement with the ground truth, for which the training pipeline is agnostic. Insights into learnability, interpretability, and generality of the trained networks have been provided as well. The results of this study can be leveraged to devise techniques for the reconstruction of small-scale structures using large-eddy simulation data.

Event-Based Distributed Set-Membership Estimation for Complex Networks Under Deception Attacks

This paper addresses the problem of event-based distributed set-membership estimation for complex networks with unknown but bounded (UBB) disturbances. To reflect the compromised data transmissions in cyber security, deception attacks are taken into consideration. Meanwhile, a novel estimation model is proposed against UBB disturbances. In order to schedule the signal transmissions between nodes and remote estimators, a novel decentralized dynamic periodic event-triggered mechanism (DPETM) with a time-varying threshold is developed for each node of the complex networks, which reduces the waste of communication resources and the complexity of computation. Thereafter, a series of distributed set-membership estimators are designed, whose parameters are explicitly determined in terms of the resolution of a particular linear matrix inequality (LMI) related to the information of the communication topology. An optimized ellipsoid estimation set is obtained by applying a recursive optimization algorithm. Finally, the simulation results are shown to demonstrate the viability of the proposed method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper is motivated by set-membership state estimation problem of complex networks in practical missions, such as military, environment, industry, etc. The set-membership estimation of complex networks provides a reliable confidence region for each system node. Event-triggered control is an effective method for the design of set-membership estimator. But the common results require the systems to monitor the measurements point-to-point, which leads to the huge consumption of calculation and communication resources. For this reason, this paper originally extends the DPETM to the discrete-time version from the field of continuous-time systems. Meanwhile, this paper considers the deception attacks in communication channels, and the generic framework established earlier can tackle simultaneously sector-bounded nonlinearity, UBB disturbances, and deception attacks. The main difficulty of this paper lies in the analysis for the sawtooth constraint of periodic samplings. For this difficulty, we introduce a piecewise auxiliary function, which is similar with the loop-function in the field of continuous-time systems. Together with recursive optimization algorithm, the detailed analysis method is proposed for the reliable confidence regions of each set-membership estimator.