Joint State Estimation Research Articles

Joint state and process inputs estimation for state-space models with Student’s t-distribution

This paper proposes a joint state and unknown inputs (UIs) discrete-time estimation method for industrial processes, represented by a state-space model. To cope with the outliers in process data, the measurement noise is characterized by the Student’s t-distribution. The identification of UIs is accomplished through the recursive expectation–maximization (REM) approach. Specifically, in the E-step, a recursively calculated Q-function is formulated by the maximum likelihood criterion, and the states and the variance scale factor are estimated iteratively. In the M-step, UIs are updated analytically together with the degree of freedom is updated approximately. The effectiveness of the proposed algorithm is validated using a quadruple water tank process and a continuous stirred tank reactor. It shows that the proposed method significantly enhances the robustness and estimation accuracy of state and UIs in industrial processes, effectively handling outliers and reducing computational demands for real-time applications.

Physics and data‐driven approach for online joint state and parameter estimation of electricity and steam networks

Physics and data‐driven approach for online joint state and parameter estimation of electricity and steam networks

Bayesian Optimization for State and Parameter Estimation of Dynamic Networks with Binary Space.

This paper focuses on joint state and parameter estimation in partially observed Boolean dynamical systems (POBDS), a hidden Markov model tailored for modeling complex networks with binary state variables. The majority of current techniques for parameter estimation rely on computationally expensive gradient-based methods, which become intractable in most practical applications with large size of networks. We propose a gradient-free approach that uses Gaussian processes to model the expensive log-likelihood function and utilizes Bayesian optimization for efficient likelihood search over parameter space. Joint state estimation is also achieved alongside parameter estimation using the Boolean Kalman filter. The performance of the proposed method is demonstrated using gene regulatory networks observed through synthetic gene-expression data. The numerical results demonstrate the scalability and effectiveness of the proposed method in the joint estimation of the model parameters and genes' states.

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.

Joint estimation of State of Charge (SOC) and State of Health (SOH) for lithium ion batteries using Support Vector Machine (SVM), Convolutional Neural Network (CNN) and Long Sort Term Memory Network (LSTM) models

This paper proposes a data-driven method for jointly estimating the State of Charge (SOC) and State of Health (SOH) of batteries, addressing the impact of battery aging on SOC estimation. Initially, Support Vector Machine (SVM) is employed to estimate the SOH of the battery, using the constant voltage charging time and constant voltage discharging time of lithium-ion batteries as inputs, and SOH as the output. By training the SVM model, accurate SOH estimation is achieved. Subsequently, the rated capacity of the battery is adjusted based on the estimated SOH to obtain the current maximum available capacity. This adjustment allows for the coupling of SOH and SOC, resulting in SOC estimation that accounts for aging factors. Leveraging the advantages of Convolutional Neural Networks (CNN) in feature extraction and Long Short-Term Memory (LSTM) neural networks in handling long-term sequential data, a CNN-LSTM model is utilized for SOC estimation. The proposed method utilizes the Oxford Battery Dataset (Cells 1–8) and the NASA Battery Dataset (B0005–B0007) for SOH estimation, and the Oxford Battery Dataset (Cell 8) and the NASA Battery Dataset (B0007) for SOC estimation. The SOH estimation results demonstrate that the Root Mean Square Error (RMSE) is less than 0.81 % and the Mean Absolute Error (MAE) is less than 0.65 % for Cells 1–8, while for B0005 to B0007, the RMSE is less than 1.81 % and the MAE is less than 1.29 %. For SOC estimation, the results show that the average RMSE over the entire lifecycle of Cell 8 is 0.3923 % and the average MAE is 0.3339 %, whereas for B0007, the average RMSE over the entire lifecycle is 0.6123 % and the average MAE is 0.4976 %.

Comparative Study of Joint Estimation of State of Charge (SOC) and State of Health (SOH) of Lithium-ion Batteries Based on Different Tree Models

The realization of accurate State of Health (SOH) and State of Charge (SOC) estimation is a prerequisite to ensure the safe use of energy storage batteries, which helps to further improve the energy utilization efficiency effectively. Data-driven methods are efficient, accurate, and do not rely on accurate battery models, which is a hot direction in battery state estimation research. However, the relationships between variables in the lithium-ion battery dataset are mostly nonlinear, which largely affects the prediction of the model. In addition, the model also has a series of defects, such as large computation, strong data dependence, and long consumption time. In this paper, a joint online estimation method of battery SOC-SOH based on tree modeling algorithm is proposed to solve the above problems. Based on NASA battery sample data, this study explores the changing law between SOC and discharge voltage and temperature under different State of Health (SOH). Subsequently, a combination of RFR, GBDT and XGBoost tree modeling algorithms are used for battery SOC-SOH estimation based on the above variation rules. The experimental results show that the R<sup>2</sup> scores of the XGBoost algorithm in predicting both SOC and SOH are more than 0.995, indicating its good adaptability and feasibility.

A New Observer Design for the Joint Estimation of States and Unknown Inputs for a Class of Nonlinear Fractional-Order Systems

This article designs an observer for the joint estimation of the state and the unknown input for a class of nonlinear fractional-order systems (FOSs) such that one portion satisfies the Lipschitz condition and the other does not necessarily satisfy such a condition. Firstly, by reconstructing system dynamics, the observer design is transformed equivalently into the tracking problem between the original nonlinear FOSs and the designed observer. Secondly, the parameterized matrices of the desired observer are derived by use of the property of the generalized inverse matrices and the linear matrix inequality (LMI) technique combined with the Schur complement lemma. Moreover, an algorithm is presented to determine the desired observer for the nonlinear FOSs effectively. Finally, a numerical example is reported to verify the correctness and efficiency of the proposed algorithm.

Distributed Energy-Based Estimation Over Harvesting-Constrained Sensor Networks.

This article investigates the distributed joint state and fault estimation issue for a class of nonlinear time-varying systems over sensor networks constrained by energy harvesting. It is assumed that data transmission between sensors requires energy consumption, and each sensor can harvest energy from the external environment. A Poisson process models the energy harvested by each sensor, and the sensor's transmission decision depends on its current energy level. One can obtain the sensor transmission probability through a recursive calculation of the probability distribution of the energy level. Under such energy harvesting constraints, the proposed estimator only uses local and neighbor data to simultaneously estimate the system state and the fault, thereby establishing a distributed estimation framework. Moreover, the estimation error covariance is determined to possess an upper bound, which is minimized by devising energy-based filtering parameters. The convergence performance of the proposed estimator is analyzed. Finally, a practical example is presented to verify the usefulness of the main results.

Particle Filter based on Jaya optimisation for Bayesian updating of nonlinear models

Particle filter (PF) is a powerful and commonly used filtering technique based on Sequential Monte Carlo framework. The main challenge in using PF for nonlinear state and parameter estimation is the degeneracy of particles. Although resampling techniques can solve this to some extent, it would still result in particle impoverishment when a limited number of particles are used thereby affecting the accuracy. Hence, a hybrid metaheuristic optimisation algorithm that combines the PF with Jaya optimisation, (PF-JAYA) has been proposed and implemented for joint state and parameter estimation for geotechnical engineering problems. The performance of PF-JAYA has been compared against the traditional Particle Filter with Sampling Importance Resampling (PF-SIR) technique. The synthetic examples show that PF-JAYA outperforms PF-SIR in terms of accuracy, rate of convergence, parameter identification and particle diversity. Furthermore, the performance of PF-JAYA is independent of the choice of prior distribution and due to its superior convergence proves to be efficient when working with sparse monitoring information. The performance of PF-JAYA on Bayesian updating of state and parameters of an elastoplastic model for a synthetic embankment case has also been evaluated where, along with PF-SIR, the Ensemble Kalman Filter (EnKF) is also chosen for comparison. Finally a further evaluation using the Lorenz ‘63 model, shows the superior performance of PF-JAYA in terms of accuracy and precision over the classical Data Assimilation techniques.

A Joint State and Fault Estimation Scheme for State-Saturated System with Energy Harvesting Sensors.

In this article, the issue of joint state and fault estimation is ironed out for delayed state-saturated systems subject to energy harvesting sensors. Under the effect of energy harvesting, the sensors can harvest energy from the external environment and consume an amount of energy when transmitting measurements to the estimator. The occurrence probability of measurement loss is computed at each instant according to the probability distribution of the energy harvesting mechanism. The main objective of the addressed problem is to construct a joint state and fault estimator where the estimation error covariance is ensured in some certain sense and the estimator gain is determined to accommodate energy harvesting sensors, state saturation, as well as time delays. By virtue of a set of matrix difference equations, the derived upper bound is minimized by parameterizing the estimator gain. In addition, the performance evaluation of the designed joint estimator is conducted by analyzing the boundedness of the estimation error in the mean-squared sense. Finally, two experimental examples are employed to illustrate the feasibility of the proposed estimation scheme.

Joint Estimation of State of Charge and State of Energy of Lithium-Ion Batteries Based on Optimized Bidirectional Gated Recurrent Neural Network

Joint Estimation of State of Charge and State of Energy of Lithium-Ion Batteries Based on Optimized Bidirectional Gated Recurrent Neural Network

Joint Estimation of State of Charge and State of Health of Lithium-Ion Batteries Based on Stacking Machine Learning Algorithm

Conducting online estimation studies of the SOH of lithium-ion batteries is indispensable for extending the cycle life of energy storage batteries. Data-driven methods are efficient, accurate, and do not depend on accurate battery models, which is an important direction for battery state estimation research. However, the relationships between variables in lithium-ion battery datasets are mostly nonlinear, and a single data-driven algorithm is susceptible to a weak generalization ability affected by the dataset itself. Meanwhile, most of the related studies on battery health estimation are offline estimation, and the inability for online estimation is also a problem to be solved. In this study, an integrated learning method based on a stacking algorithm is proposed. In this study, the end voltage and discharge temperature were selected as the characteristics based on the sample data of NASA batteries, and the B0005 battery was used as the training set. After training on the dataset and parameter optimization using a Bayesian algorithm, the trained model was used to predict the SOH of B0007 and B0018 models. After comparative analysis, it was found that the prediction results obtained based on the proposed model not only have high accuracy and a short running time, but also have a strong generalization ability, which has a great potential to achieve online estimation.

Distributed joint parameter and state estimation algorithm for large‐scale interconnected systems

SummaryThis paper proposes a distributed joint parameter and state variables estimation algorithm for large‐scale state‐space interconnected systems. In this distributed estimation scheme, each interconnected sub‐system is described by a linear discrete‐time state space mathematical model. Each sub‐system is supposed to be controlled by an intelligent controller that can communicate with its interconnected neighbors and exchange information, such as state variables. The proposed approach comprises two recursive estimation algorithms, a parameter estimation algorithm considering the state space model and a distributed Kalman filter for state variables estimation. It is a fully distributed cooperative approach that allows to reduce complexity and saves computational and communication resources. Theoretical analysis and numerical examples are provided to prove the feasibility and effectiveness of this joint estimation algorithm.

How Not to Make the Joint Extended Kalman Filter Fail with Unstructured Mechanistic Models.

The unstructured mechanistic model (UMM) allows for modeling the macro-scale of a phenomenon without known mechanisms. This is extremely useful in biomanufacturing because using the UMM for the joint estimation of states and parameters with an extended Kalman filter (JEKF) can enable the real-time monitoring of bioprocesses with unknown mechanisms. However, the UMM commonly used in biomanufacturing contains ordinary differential equations (ODEs) with unshared parameters, weak variables, and weak terms. When such a UMM is coupled with an initial state error covariance matrix P(t=0) and a process error covariance matrix Q with uncorrelated elements, along with just one measured state variable, the joint extended Kalman filter (JEKF) fails to estimate the unshared parameters and state simultaneously. This is because the Kalman gain corresponding to the unshared parameter remains constant and equal to zero. In this work, we formally describe this failure case, present the proof of JEKF failure, and propose an approach called SANTO to side-step this failure case. The SANTO approach consists of adding a quantity to the state error covariance between the measured state variable and unshared parameter in the initial P(t = 0) of the matrix Ricatti differential equation to compute the predicted error covariance matrix of the state and prevent the Kalman gain from being zero. Our empirical evaluations using synthetic and real datasets reveal significant improvements: SANTO achieved a reduction in root-mean-square percentage error (RMSPE) of up to approximately 17% compared to the classical JEKF, indicating a substantial enhancement in estimation accuracy.

Joint State and Unknown Input Estimation for a Class of Artificial Neural Networks With Sensor Resolution: An Encoding-Decoding Mechanism.

This article is concerned with the joint state and unknown input (SUI) estimation for a class of artificial neural networks (ANNs) with sensor resolution (SR) under the encoding-decoding mechanisms. The consideration of SR, which is an important specification of sensors in the real world, caters to engineering practice. Furthermore, the implementation of the encoding-decoding mechanism in the communication network aims to accommodate the limited bandwidth. The objective of this study is to propose a set-membership estimation algorithm that accurately estimates the state of the ANN without being influenced by the unknown input while accounting for the SR and the encoding-decoding mechanism. First, a sufficient condition is derived to ensure an ellipsoidal constraint on the estimation error. Then, by addressing an optimization problem, the design of the estimator gains is accomplished, and the minimal ellipsoidal constraint on the state estimation error is obtained. Finally, an example is provided to confirm the validity of the proposed joint SUI estimation scheme.

Sequential Joint State Estimation and Track Extraction Algorithm Based on Improved Backward Smoothing

In order to further promote the accuracy of state estimation and track extraction capabilities for multi-target tracking, a sequential joint state estimation and track extraction algorithm is proposed in this article. This algorithm is based on backward smoothing under the framework of Labeled Random Finite Set and utilizes label iterative processing to perform outlier removal, invalid short-lived track removal, and track continuity processing in a sequential manner in order to achieve the goal of improving the multi-target tracking performance of the algorithm. Finally, this paper verifies through experimental simulation and measured data analysis that the proposed algorithm has improved the performance of radar multi-target tracking to a certain extent.

Robust Bayesian state and parameter estimation framework for stochastic dynamical systems with combined time-varying and time-invariant parameters

We consider the joint state and parameter estimation for dynamical systems having both time-varying and time-invariant parameters. For systems with time-invariant parameters, it has previously been demonstrated that although nonlinear filters may be used for concurrent state and parameter estimation, employing the Markov Chain Monte Carlo (MCMC) algorithm for estimating time-invariant parameters with nested nonlinear filters for state estimation provides more reliable estimates. Following established methods for parameter estimation using filters, we augment the state vector to include the original system states in addition to the subset of the parameters that are known to vary in time. Conventionally, both time-varying and time-invariant parameters would be appended in the state vector, and thus for the purpose of estimation, both would free to vary in time. However, allowing parameters that are known to be time-invariant to change in time introduces non-physical dynamics into the original system. To avoid inducing artificial dynamics, the time-invariant parameters are estimated using MCMC. Furthermore, by estimating the time-invariant parameters by MCMC, the dimensionality of the augmented state is smaller and the nonlinearity in the ensuing state space model will tend to be weaker compared to the conventional approach. The time-varying parameters are perturbed by random noise processes, and the amplitude of the artificial noise is estimated alongside the time-invariant system parameters using MCMC. Estimating the noise strength in this procedure circumvents the need for manual tuning. We illustrate the above-described approach for a simple dynamical system in which some model parameters are time-varying, while the remaining parameters are time-invariant. We adopt the extended Kalman filter (EKF) for state estimation and the estimation of the time-varying system parameters, but reserve the task of estimating time-invariant parameters, including the model noise strength and the artificial noise strength associated with the time-varying parameter to the MCMC algorithm.

Vehicle State Joint Estimation Based on Lateral Stiffness.

In this study, a vehicle state joint estimation method based on lateral stiffness was applied to estimate the running states of electric vehicles driven by rear-drive, in-wheel motors. Different from the estimation methods used in other research, the joint estimator designed in this study uses the least-squares (LS) algorithm to estimate the lateral stiffness of the front and rear axles of the vehicle, deploying the high-degree cubature Kalman filter algorithm to estimate the vehicle state. We establish a three-degree-of-freedom nonlinear vehicle model with longitudinal velocity, lateral velocity, and yaw rate, and the lateral stiffness of the front and rear axles as the principal parameters. For the low-speed running state of the vehicle, a linearized magic tire model with high fitting accuracy was used to calculate the lateral force of the entire vehicle. The LS algorithm with a forgetting factor was used to design a lateral stiffness estimator to assess the front-axle and rear-axle lateral stiffness of the entire vehicle. The generalized high-degree cubature Kalman filter (GHCKF) algorithm was used to design the vehicle state estimator and further improve the GHCKF algorithm. A vehicle state estimator, using the square root generalized high-degree cubature Kalman filter (SRGHCKF), was designed. Therefore, the joint estimator, comprising a lateral stiffness estimator and a vehicle state estimator, adopts the LS-GHCKF/SRGHCKF algorithm and enables the estimation of the lateral stiffness, the longitudinal velocity, the lateral velocity, and the yaw rate of the entire vehicle during the driving process. A double lane change and slalom simulation were performed to analyze the feasibility and accuracy of the joint estimation algorithm and verify the results of the LS-GHCKF algorithm and the LS-SRGHCKF algorithm. Further, a low-speed driving experiment was carried out for electric vehicles driven by rear in-wheel motors. The inertial navigation system (INS), the global positioning system (GPS), the real-time kinematic (RTK), and an angle sensor were used to collect real-time vehicle data. The results were compared to verify the feasibility of the joint estimator and the progressiveness of the algorithm. The experimental verification and simulation both show that the vehicle state joint estimator, designed based on the LS-GHCKF/SRGHCKF algorithm, can accurately estimate the real-time state of the vehicle. Additionally, the LS-SRGHCKF algorithm shows better effectiveness and robustness than the LS-GHCKF algorithm.

Adaptive Fuzzy Boundary Observer Design for Uncertain Linear Coupled Hyperbolic Partial Differential Equation Systems

Joint uncertainties and state estimation of a class of linear coupled hyperbolic partial differential equation systems in the presence of unstructured and structured uncertainties are studied in this paper. For unstructured uncertainties which are completely unknown, by employing Takagi-Sugeno fuzzy logic system to approximate the unstructured uncertainties, a novel adaptive fuzzy boundary observer is developed to estimate both unknown system states as well as unknown weights in the fuzzy logic system, and the estimation errors are ultimately bounded. Therein, in the design of the proposed observer, a set of swapping filters and infinite dimensional backstepping technique are combined. On the other hand, for structured uncertainties that can be described in a concrete parameterized form, the proposed method can easily achieve the exact estimation of weights and states to their true values. The rigorous proof is provided to show that the ultimately bounded estimation errors for the case of unstructured uncertainties and the exponential convergent estimation errors for the case of structured uncertainties can be realized. Finally, three illustrative simulations are carried out to show the feasibility and effectiveness of the developed methods in this paper.

Joint Topology and State Estimation Using TTU Monitoring Data Considering Three-Phase Imbalance

A grand challenge for operators to develop optimal controls at the distribution level lies in how to obtain the real-time network topology and operation state in lack of specialized high precision monitoring devices, especially with high penetration of distributed renewable energy. Driven by this motivation, this paper proposes a novel topology and operation state joint estimation framework based on the monitoring data from transformer terminal units (TTUs). In the framework, the bottom layer generates an equivalent network model and nodal voltage and current measurements with synchronized phase angles from the distribution network and the voltage and power measurement data from TTU. The middle layer formulates, linearizes and solves an optimization problem to fit the model with the measurements. A global piecewise linearization method is proposed to convert the original mixed integer quadratically constrained program (MIQCP) problem into a mixed integer linear program (MILP) problem. In the top layer, the line switch states and feeder power flows are estimated with the solution of the optimization problem. Case studies are conducted in test systems to verify the accuracy, efficiency and robustness of the proposed framework and show that the proposed framework yields obvious advantages over the existing approaches on the estimation accuracy when using TTU measurements.