Nonlinear State Estimation Research Articles

Hierarchical fault propagation path recognition method based on knowledge-driven graph attention autoencoder with bilayer pooling for large-scale industrial system

Large-scale industrial system exhibits hierarchical structure, which consists of multiple subsystem blocks with various corresponding variables. However, existing fault propagation path recognition methods fail to fully leverage the relevant knowledge about industrial system with hierarchal structure, which makes the causality inference result inconsistent with reality, causing inaccurate fault propagation path recognition. In this work, a hierarchical fault propagation path recognition method based on knowledge-driven graph attention autoencoder with bilayer pooling for large-scale industrial system is proposed. First, a knowledge-driven graph attention autoencoder is developed for causality knowledge exploration on intra-subsystem level, which expressly represents the causalities between operation variables in each subsystem into a weighted directed graph. Then, a graph bilayer pooling model is designed to generate the subsystem causality graph by “stacking” the intra-subsystem causality graphs in a hierarchical fashion. Utilizing the knowledge-driven graph attention autoencoder with bilayer pooling, the critical hierarchical characteristics of system causality can be well described. Moreover, to reduce the smearing effect, a nonlinear probabilistic state estimation model with fault isolation is adopted for distributed fault detection. Eventually, the performance of the proposed method is verified on large-scale hierarchical industrial system air separation unit with three fault cases. The fault detection and fault root cause location results show that compared with other methods, the proposed method achieves high fault detection rate with low fault alarm rate, as well as better fault root cause location performance.

Distributed minimum error entropy with fiducial points Kalman filter for state tracking

With the growing size of the system, this distributed Kalman filter (DKF) is widely used in multi-sensor networks. However, it is difficult for DKF to accurately estimate state values in non-Gaussian noise environments. In this paper, a regression equation is first constructed to contain all sensor node information. Then, by bringing the minimum error entropy with fiducial points (MEEF) standard into the process of information fusion, a robust algorithm named centralized MEEF KF (CMEEF-KF) is presented, which is robust to non-Gaussian noise and unusual data. Furthermore, to overcome the communication burden of CMEEF-KF in sensor networks, the distributed MEEF-KF (DMEEF-KF) is developed, which construct a framework of consensus average method for node information fusion. Specifically, each sensor only exchanges the key information with its neighborhoods. In addition, in order to make the algorithm able to cope with the nonlinear state estimation problem, the distributed MEEF extended Kalman filter is also proposed. Eventually, the effectiveness of the suggested algorithms is demonstrated by land vehicle navigation and power system tracking state estimation using a 10-node sensor network.

Event-triggered nonlinear state estimation with quantized innovations

This paper proposes an event-triggered (ET) estimator with quantization mechanism to relieve the bandwidth and energy limitations of wireless sensors, where the ET strategy reduces the overall communication rate of the system, and the quantization mechanism mitigates the instantaneous bit rate constraint. Then, Gaussian approximation is applied to derive the analytical form of the minimum mean squared error (MMSE) estimator under the event-triggered strategy and quantization mechanism. Furthermore, the relationship between the event-triggered parameter and the communication rate is explicitly established. Finally, theoretical results are validated through some simulations on a target tracking system.

Neural-Rendezvous: Provably Robust Guidance and Control to Encounter Interstellar Objects

Interstellar objects (ISOs) are likely representatives of primitive materials invaluable in understanding exoplanetary star systems. Due to their poorly constrained orbits with generally high inclinations and relative velocities, however, exploring ISOs with conventional human-in-the-loop approaches is significantly challenging. This paper presents Neural-Rendezvous—a deep-learning-based guidance and control framework for encountering fast-moving objects, including ISOs, robustly, accurately, and autonomously in real time. It uses pointwise minimum norm tracking control on top of a guidance policy modeled by a spectrally normalized deep neural network, where its hyperparameters are tuned with a loss function directly penalizing the model predictive control state trajectory tracking error. We show that Neural-Rendezvous provides a high probability exponential bound on the expected spacecraft delivery error, the proof of which leverages stochastic incremental stability analysis. In particular, it is used to construct a non-negative function with a supermartingale property, explicitly accounting for the ISO state uncertainty and the local nature of nonlinear state estimation guarantees. In numerical simulations, Neural-Rendezvous is demonstrated to satisfy the expected error bound for 100 ISO candidates. This performance is also empirically validated using our spacecraft simulator and in high-conflict and distributed unmanned aerial vehicle swarm reconfiguration with up to 20 unmanned aerial vehicles.

Intelligent Monitoring and Optimization of Wind Turbine Operation Status on Basis of Vibration Signals

In response to the large monitoring errors and incomplete consideration of fault locations in the operation status of wind turbines, this paper combined vibration signals and used a model constructed using nonlinear state estimation technology (NSET) to monitor and optimize the operation status of wind turbines. Firstly, it collected relevant SCADA (supervisory control and data acquisition) and vibration sensor data of Unit 5 of Guangdong Yangjiang Shaba Offshore Wind Farm on site, and used wavelet transform to denoise the vibration signal. Then, full consideration can be given to the faulty parts such as wind blades and gearboxes, and a model constructed by NSET using Markov distance to construct a process memory matrix can be used to monitor the status of the wind turbine equipment. Finally, the multi-level fuzzy comprehensive evaluation method can be used to optimize the comprehensive evaluation of the operating status of wind turbines, reducing monitoring errors.

Iterative UKF under generalized maximum correntropy criterion for intermittent observation systems with complex non-Gaussian noise

The traditional unscented Kalman filters (UKFs) under the maximum correntropy criterion provide a powerful tool for nonlinear state estimation with heavy-tailed non-Gaussian noise. Nevertheless, the above-mentioned filters may yield biased estimates because the Gaussian kernel function can only handle certain types of non-Gaussian noise. Additionally, the use of statistical linearization methods can result in approximation errors when solving linear observation equations, while the system may also experience observation data loss. Therefore, a new iterative UKF with intermittent observations under the generalized maximum correntropy criterion is proposed for systems with complex non-Gaussian noise, called GMCC-IO-IUKF. Firstly, the connection between the UKF with and without intermittent observations is established by designing a coefficient matrix including intermittent observation variables, so as to derive the UKF with intermittent observations under the maximum correntropy criterion. Secondly, for the measurement update of GMCC-IO-IUKF, a nonlinear regression augmented model that can deal with both prediction and observation errors is established using the coefficient matrix and the nonlinear function. To better adapt to different types of non-Gaussian noise, the generalized Gaussian kernel function is substituted for the traditional Gaussian kernel function. Theoretically, GMCC-IO-IUKF can achieve better estimation performance by directly employing the nonlinear function and the latest iteration value. Finally, a classical target tracking model is used to evaluate the estimation performance and feasibility of our proposed GMCC-IO-IUKF algorithm. It appears from the experiment results that our proposed GMCC-IO-IUKF can not only promote estimation precision but also handle complex non-Gaussian noise flexibly.

Robust recursive sigma point Kalman filtering for Huber-based generalized M-estimation

For nonlinear state estimation driven by non-Gaussian noise, the estimator is required to be updated iteratively. Since the iterative update approximates a linear process, it fails to capture the nonlinearity of observation models, and this further degrades filtering accuracy and consistency. Given the flaws of nonlinear iteration, this work incorporates a recursive strategy into generalized M-estimation rather than the iterative strategy. The proposed algorithm extends nonlinear recursion to nonlinear systems using the statistical linear regression method. The recursion allows for the gradual release of observation information and consequently enables the update to proceed along the nonlinear direction. Considering the correlated state and observation noise induced by recursions, a separately reweighting strategy is adopted to build a robust nonlinear system. Analogous to the nonlinear recursion, a robust nonlinear recursive update strategy is proposed, where the associated covariances and the observation noise statistics are updated recursively to ensure the consistency of observation noise statistics, thereby completing the nonlinear solution of the robust system. Compared with the iterative update strategies under non-Gaussian observation noise, the recursive update strategy can facilitate the estimator to achieve higher filtering accuracy, stronger robustness, and better consistency. Therefore, the proposed strategy is more suitable for the robust nonlinear filtering framework.

Adaptive polynomial Kalman filter for nonlinear state estimation in modified AR time series with fixed coefficients

AbstractThis article presents a novel approach for adaptive nonlinear state estimation in a modified autoregressive time series with fixed coefficients, leveraging an adaptive polynomial Kalman filter (APKF). The proposed APKF dynamically adjusts the evolving system dynamics by selecting an appropriate autoregressive time‐series model corresponding to the optimal polynomial order, based on the minimum residual error. This dynamic selection enhances the robustness of the state estimation process, ensuring accurate predictions, even in the presence of varying system complexities and noise. The proposed methodology involves predicting the next state using polynomial extrapolation. Extensive simulations were conducted to validate the performance of the APKF, demonstrating its superiority in accurately estimating the true system state compared with traditional Kalman filtering methods. The root‐mean‐square error was evaluated for various combinations of standard deviations of sensor noise and process noise for different sample sizes. On average, the root‐mean‐square error value, which represents the disparity between the true sensor reading and estimate derived from the adaptive Kalman filter, was 35.31% more accurate than that of the traditional Kalman filter. The comparative analysis highlights the efficacy of the APKF, showing significant improvements in state estimation accuracy and noise resilience.

A new observer-based controller for continuous-time nonlinear systems with applications on electrical energy systems

In this paper a new observer-based controller is developed for continuous-time input constrained nonlinear systems. Firstly, a new developed continuous time Regularized Least Square (CRLS) estimator is presented. The estimated states are then used to generate the desired constrained state feedback control strategies to regulate the performance of a perturbed system to the desired steady state. Unlike the well-known continuous-time nonlinear state estimators, such as the high gain observer and others, the developed observer does not show large overshoots at the start of the estimation process, the estimation errors reach zero steady state very shortly, has no pre-specified limitations on the class of nonlinear systems to be estimated, …etc. As a result, when used as an element in observer-based controlled systems, then the desired behaviors of the state trajectories of the controlled system are reached very shortly. The asymptotic stability of the proposed constrained observer-based controlled nonlinear system is rigorously analyzed. Simulation results are presented to justify the applicability and the efficiency of the proposed approach. Firstly, to show the superiority of the developed CRLS estimator, the states of illustrative practical problems are firstly estimated using the proposed approach. The achieved results are then compared with those achieved from applying the most famous and widely used observers in the literature. Then, two nonlinear practical control problems are used to demonstrate the effectiveness of the developed observer- based control approach.

Improved Particle Filter Algorithm for Multi-Target Detection and Tracking.

In modern radar detection systems, the particle filter technique has become one of the core algorithms for real-time target detection and tracking due to its good nonlinear and non-Gaussian system state estimation capability. However, when dealing with complex dynamic scenes, the traditional particle filter algorithm exposes obvious deficiencies. The main expression is that the sample degradation is serious, which leads to a decrease in estimation accuracy. In multi-target states, the algorithm is difficult to effectively distinguish and stably track each target, which increases the difficulty of state estimation. These problems limit the application potential of particle filter technology in multi-target complex environments, and there is an urgent need to develop a more advanced algorithmic framework to enhance its robustness and accuracy in complex scenes. Therefore, this paper proposes an improved particle filter algorithm for multi-target detection and tracking. Firstly, the particles are divided into tracking particles and searching particles. The tracking particles are used to maintain and update the trajectory information of the target, and the searching particles are used to identify and screen out multiple potential targets in the environment, to sufficiently improve the diversity of the particles. Secondly, the density-based spatial clustering of applications with noise is integrated into the resampling phase to improve the efficiency and accuracy of particle replication, so that the algorithm can effectively track multiple targets. Experimental result shows that the proposed algorithm can effectively improve the detection probability, and it has a lower root mean square error (RMSE) and a stronger adaptability to multi-target situation.

Application of fixed-lag Kalman smoother for nonlinear state estimation

Application of fixed-lag Kalman smoother for nonlinear state estimation

Nonlinear state estimation of a power plant superheater by using the extended Kalman filter for differential algebraic equation systems

A nonlinear estimator was developed for process monitoring of a power plant superheater system to estimate the transient temperature profiles of the critical measured and unmeasured variables under load-following operations. The model used in the estimator was a detailed, distributed, first-principles, differential–algebraic equation model. The superheater geometry is characterized based on an operating natural gas combined cycle power plant. The model included mass and energy balances for the operating fluids, i.e., steam and flue gas, with due consideration of tube wall temperature dynamics. The dynamic model was used for state and parameter estimation by using a modified extended Kalman filter. When results are compared with the data from a natural gas power plant, the estimator yielded root mean squared error of 0.2 °C for the main steam temperature compared to 1 °C obtained from the first-principles model. While the first-principles model yielded as high as 2.5 °C absolute instantaneous error for the tube temperatures, the estimator reduced the absolute instantaneous error below 0.3 °C for most variables. Performance of the estimator for estimating critical variables like the main steam temperature and flue gas temperature was evaluated in the presence of high model mismatch and noisy measurement data by simulating load-following operation of the plant. The estimator yielded maximum absolute instantaneous error of 3.4 °C for the flue gas and 1 °C for the main steam temperature.

Estimation of machinery’s remaining useful life in the presence of non-Gaussian noise by using a robust extended Kalman filter

Estimation of the remaining useful life (RUL) of industrial machinery is essential for condition-based maintenance (CBM). While numerous papers have explored this issues, challenges arise as machinery often works in non-stationary conditions, particularly in harsh environments (like mining machines, wind turbines, helicopters, etc.). The data collected from such environments are affected by non-Gaussian noise, posing difficulties for traditional approaches to non-linear state estimation or prediction. The widely used extended Kalman filter (EKF) suffers from the non-Gaussian noise effect due to its recursive minimum L2-norm filtering. To address these issues, we propose a robust EKF based on the maximum correntropy criterion. This method effectively estimates the RUL of the time-varying degradation process in the presence of non-Gaussian noise, also enabling confidence interval computation for uncertainty management. The efficiency of our approach was confirmed through application to simulated and benchmark data sets, outperforming Kalman filter-based methods for both simulated and real-world scenarios.

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.

Learning-Based Faulty State Estimation Using SOH-Coupled Model Under Internal Thermal Faults in Lithium-Ion Batteries

Estimating the state of charge (SOC), state of health (SOH), and core temperature under internal faults will significantly improve the battery management system’s (BMS’s) autonomy and accuracy in range prediction. This paper presents a neural network (NN) based state estimation scheme that can estimate the SOC, core temperature, and SOH under internal faults in lithium-ion batteries (LIBs). First, we propose a model-based internal fault detection scheme by employing a SOH-coupled electro-thermal-aging model (ETA) of the LIB. Then, a nonlinear observer is used to estimate the proposed SOH-coupled model’s healthy states for a residual generation. The fault diagnosis scheme compares the output voltage and surface temperature residuals against the designed adaptive threshold to detect thermal faults. The adaptive threshold effectively alleviates the false positives due to degradation and model uncertainties of the battery under no-fault conditions. Upon fault detection, we employ an additional NN-based observer in the second step to learn the faulty dynamics. A novel NN weight tuning algorithm is proposed using the measured voltage, surface temperature, and estimated healthy states. The convergence of the nonlinear and NN-based observer state estimation errors is proven using the Lyapunov theory. Finally, numerical simulation results are presented.

False Data Injection Attack with Max-Min Optimization in Smart Grid

With the proliferation of information and communication technology (ICT), the smart grid is critically vulnerable to cyber-attacks such as false data injection (FDI), denial-of-service, and data spoofing. The cyber-attackers defunctionalize critical operations of the smart grid by compromising the ICT. The decisions for the critical operation of the smart grid are processed with a state estimator, and ICT makes the state estimator vulnerable to FDI attacks. Hence, vulnerability analysis of the state estimator needs to be investigated for potential FDI attacks to protect it from future cyber-attacks. This work proposes the FDI attack vector construction without the knowledge of bad data detection (BDD) threshold on linear and non-linear state estimators using max-min optimization considering the partial network information. The optimization problem is formulated from the attacker's perspective to target the manipulation of measurements in the attack zone so as to increase the generation cost. The equivalent power injection model is developed for the attack zone to construct the deceptive attacks using DC and AC power flow models. The effectiveness of the proposed framework has been tested on 5 bus PJM network, modified IEEE 30 bus, IEEE 57 bus, and IEEE 118 bus system. The proposed framework is compared with existing state-of-art methods (viz., linear attack policy, non-linear attack policy, load redistribution attack, and line flow attack) to assess its efficacy. It is observed that the developed FDI attack vector successfully bypasses the bad data detectors such as the chi-square test, largest normalized residue, and l2 detector that is normally used by the system operator. Moreover, the study compares and discusses the impact of FDI attacks on the estimated state variables obtained with state estimators and, thereby, the generation cost calculated with the DC & AC optimal power flow tool.

A novel fractional nonlinear state estimation algorithm in non-Gaussian noise environment

This paper presents research on state estimation for nonlinear fractional-order systems in non-Gaussian noise. Traditional estimation methods based on the minimum mean square error criterion perform poorly for fractional-order systems in non-Gaussian noise Using the Bayesian filtering framework and information theoretic learning, a novel fractional central difference Kalman filter based on the maximum correntropy criterion is firstly proposed and named MC-FCDKF. The proposed MC-FCDKF contains prior state update step, regression model construction and posterior state update step. The regular Taylor series expansion is replaced by the Stirling interpolation technique. The conventional measurement prediction and the associated calculation of covariances are also eliminated, enhancing practicality and real-time performance. Finally, the performance of the proposed algorithm is compared with several other algorithms through lithium battery state-of-charge estimation to verify the effectiveness and benefits under non-Gaussian noise perturbation.

A deep learning technique to control the non-linear dynamics of a gravitational-wave interferometer

Abstract In this work we developed a deep learning technique that successfully solves a non-linear dynamic control problem. Instead of directly tackling the control problem, we combined methods in probabilistic neural networks and a Kalman-filter-inspired model to build a non-linear state estimator for the system. We then used the estimated states to implement a trivial controller for the now fully observable system. We applied this technique to a crucial non-linear control problem that arises in the operation of the Laser Interferometer Gravitational-Wave Observatory (LIGO) system, an interferometric gravitational-wave observatory. We demonstrated in simulation that our approach can learn from data to estimate the state of the system, allowing a successful control of the interferometer’s mirror. We also developed a computationally efficient model that can run in real time at high sampling rate on a single modern CPU core, one of the key requirements for the implementation of our solution in the LIGO digital control system. We believe these techniques could be used to help tackle similar non-linear control problems in other applications.

Early Prediction of Remaining Useful Life for Rolling Bearings Based on Envelope Spectral Indicator and Bayesian Filter

On top of the condition-based maintenance (CBM) practice for rotating machinery, the robust estimation of remaining useful life (RUL) for rolling-element bearings (REB) is of particular interest. The failure of a single bearing often results in secondary defects in the connected structure and catastrophic system failures. The prediction of RUL facilitates proactive maintenance planning to ensure system reliability and minimize financial loss due to unscheduled downtime. In this paper, to acquire early and reliable estimations of useful life, the RUL prediction of REBs is formulated into nonlinear degradation state estimation tackled by the combination of the envelope spectral indicator (ESI) and extended Kalman filter (EKF). By fusing the spectral energy of the bearing fault characteristic frequencies (FCFs) in the averaged envelope spectrum, the ESI is crafted to remove the interference from rotor-dynamics and reveal the bearing deterioration process. Once the fault is identified, the recursive Bayesian method based on EKF is utilized for estimating the bearing end-of-life time via the exponential state-space model. The distinctive advantage of the proposed approach lies in its ability to make an early prediction of RUL using a small number of ESI observations, offering an efficient practice for predictive health management at the early stage of bearing fault. The performance of the proposed method is validated using publicly available experimental bearing vibration data across three different operating conditions.

In-context learning of state estimators

State estimation has a pivotal role in several applications, including but not limited to advanced control design. Especially when dealing with nonlinear systems state estimation is a nontrivial task, often entailing approximations and challenging fine-tuning phases. In this work, we propose to overcome these challenges by formulating an in-context state-estimation problem, enabling us to learn a state estimator for a class of (nonlinear) systems abstracting from particular instances of the state seen during training. To this end, we extend an in-context learning framework recently proposed for system identification, showing via a benchmark numerical example that this approach allows us to (i) use training data directly for the design of the state estimator, (ii) not requiring extensive fine-tuning procedures, while (iii) achieving superior performance compared to state-of-the-art benchmarks.