Measurement Noise Distribution Research Articles

Dynamic data reconciliation for enhancing the prediction performance of long short-term memory network

Abstract In modern process industries, long short-term memory (LSTM) network is widely used for data-driven modeling. Constrained by measuring instruments and environments, the measured datasets are generally with Gaussian/non-Gaussian distributed measurement noise. The noisy datasets will impact the modeling accuracy of the LSTM network and decrease the prediction performance of it. Aiming at addressing prediction performance impairment of the LSTM network under noisy datasets with Gaussian/non-Gaussian distribution, this study introduces dynamic data reconciliation (DDR) both into LSTM network training and into LSTM network test. Results show that DDR improves not only the data quality based on noisy datasets and the training outputs via the Bayesian formula in the model training step, but also the prediction performance based on offline measured information and the test outputs. The implementation scheme of DDR for Gaussian and non-Gaussian distributed noise is purposely designed. The effectiveness of DDR on the LSTM model is verified in a numerical example and a case involving a set of shared wind power datasets.

Enhancement of DDST-MFAC for tracking performance by using dynamic data reconciliation

Abstract Model-free adaptive control (MFAC) stands out as an effective data-driven method for addressing nonlinear problems in industrial processes. To maintain good control performance, a data-driven set-point tuning (DDST) method is used to update the virtual set-point of the MFAC system. The DDST-based MFAC (i.e. DDST-MFAC) constantly approaches the target of the process through the nonlinear set-point tuning method. However, due to equipment errors and external interference, industrial sensors often suffer from measurement noise, which can have adverse effects on the control performance. In this article, an available dynamic data reconciliation (DDR) technique is adopted to improve the tracking performance of the DDST-MFAC, which suppresses the impact of process noise by using predicted information and measured data to achieve high-precision requirements for controlling nonlinear processes. Finally, considering both Gaussian and non-Gaussian distribution of measurement noise, the effectiveness of the proposed method was verified through the simulation of a nonlinear nonaffine plant. It is also applied to the steam-water heat exchange process, the control result is improved ultimately.

A quantitative comparison study for structural flexibility identification using Accelerometric and computer vision-based vibration data

For civil engineering structures, the dynamics-based structural health monitoring is commonly based on inverse analysis of structural acceleration response data to identify the structural parameters. As the measurement of structural displacement response becomes more feasible and cost-effective thanks to advances in computer vision techniques, a comparison of the displacement and acceleration response data for structural identification has become of considerable interest among researchers. This paper presents a comprehensive analysis and comparison of structural flexibility identification based on displacement versus acceleration response data through uncertainty quantification technique. Firstly, theoretical derivation reveals that the energy spectral density of acceleration response is equal to the fourth power of the frequency of the energy spectral density of displacement response, which affects the distribution of measurement noise across different frequency bands. Secondly, the uncertainty quantification method for modal flexibility from subspace-based system identification is utilized to compare the derivation process of covariance estimation on modal flexibility. Finally, numerical analysis and experimental example are carried out to compare the structural flexibility identification results when using the displacement versus acceleration. It shows that more precise identification results of lower-order modal parameters are obtained when using displacement, which further improves the precision of modal flexibility identification. The displacement is more suitable for identifying the modal flexibility when the measurement noises are at the same level.

A Distributed Robust System-Wide State Estimation Method for Power Systems Based on Maximum Correntropy

The distribution of measurement noise applied in practical power system state estimation (PSSE) can deviate from the assumed Gaussian model and the performance of an estimator becomes bad if the Gaussian model is still used. This paper proposes a new distributed robust PSSE method for multi-area power systems. The non-Gaussian model is utilized to fit the measurement noise distribution to reach high model accuracy. The proposed distributed method is derived based on the maximum correntropy criterion to further reduce the impact of non-Gaussian measurement noise and outliers. The influence function combined with the finite-time average consensus algorithm is used to implement the proposed distributed robust method in a fully distributed manner. Simulations conducted on the IEEE 30-bus, 118-bus and 300-bus systems and the Polish 2383-bus system demonstrate the robustness and effectiveness of the proposed distributed method. Each local area can get the system-wide robust state estimation solution by only using local information and small amounts of data from neighboring areas. Our proposed distributed robust method has at least twelve percent improvement in reducing the mean squared error under Gaussian-Uniform noise. It is verified the communication network applied for the proposed method is fairly flexible while the existing distributed approaches use a fixed communication network when the power systems are partitioned. The simulation results also verify the robustness of the proposed method to bad data and communication failure.

A robust lp‐norm localization of moving targets in distributed multiple‐input multiple‐output radar with measurement outliers

AbstractThe Gaussian noise model and estimators based on least squares (LS) are widely used in target localisation with distributed multiple‐input multiple‐output (MIMO) radar because of their computational efficiency. However, the accuracy of existing LS‐based target localisation algorithms deteriorates sharply in the presence of outliers in the measurements. Thus, a robust solution is developed based on the ‐norm minimisation criterion and iteratively reweighted least squares (IRLS) for locating a moving target with impulse noise using the angle of arrival (AOA), time delay (TD), and Doppler shift (DS) measurements. First, the AOA, TD, and DS measurement noise models are developed based on the α‐stable distribution. Then, the localisation problem is transformed into an ‐norm minimisation problem by linearising the AOA, TD, and DS measurement equations. Finally, the ‐norm minimisation problem is solved using an IRLS method to obtain the target position and estimate the velocity. Moreover, the optimum of the norm order (p) and the Cramér–Rao lower bound for the target position and velocity estimation are derived under α‐stable distributed measurement noise. The simulation results demonstrate that the developed algorithm offers higher accurascy and robustness than the existing ones in the presence of measurement outliers.

Efficient Sensor Node Selection for Observability Gramian Optimization.

Optimization approaches that determine sensitive sensor nodes in a large-scale, linear time-invariant, and discrete-time dynamical system are examined under the assumption of independent and identically distributed measurement noise. This study offers two novel selection algorithms, namely an approximate convex relaxation method with the Newton method and a gradient greedy method, and confirms the performance of the selection methods, including a convex relaxation method with semidefinite programming (SDP) and a pure greedy optimization method proposed in the previous studies. The matrix determinant of the observability Gramian was employed for the evaluations of the sensor subsets, while its gradient and Hessian were derived for the proposed methods. In the demonstration using numerical and real-world examples, the proposed approximate greedy method showed superiority in the run time when the sensor numbers were roughly the same as the dimensions of the latent system. The relaxation method with SDP is confirmed to be the most reasonable approach for a system with randomly generated matrices of higher dimensions. However, the degradation of the optimization results was also confirmed in the case of real-world datasets, while the pure greedy selection obtained the most stable optimization results.

Dynamic data reconciliation to enhance the performance of model free adaptive control

As a novel data-driven control method, model-free adaptive control (MFAC) has a high requirement for the accuracy of the feedback signal. However, the sensor used to obtain the output inevitably contains measurement noise due to its own error or external interference, which may lead to an adverse effect on the control performance. The dynamic data reconciliation (DDR) is proposed and combined with MFAC to improve the control performance in this paper, which uses predicted output and measured data to suppress measurement noise considering Gaussian and non-Gaussian distributed measurement noise. The effectiveness of the DDR combined with MFAC (DDR-MFAC) is illustrated in the single-input single-output and multiple-input multiple-output systems with Gaussian and non-Gaussian distributed measurement noise. DDR-MFAC is also successfully applied to DC–AC converter, which improves its conversion precision.

복수 탐지정보의 공간분포 사전지식을 활용한 고해상도 차량용 레이더 표적추적 필터 설계

This paper deals with the problem of target tracking using high-resolution automotive radars. Due to the enhanced angular resolution, a high-resolution radar allows a single target to produce multiple detections originated from the main scatters with complicated spatial distribution Thus, the target tracking problem is formulated as a nonlinear state estimation associated with the multi-modal measurement noise distribution. As a practical resolution to this problem, the statistical properties of the multi-detection is modeled by approximating the reflected target signal strength as Gaussian mixture, then a Gaussian sum filter is designed using the prior knowledge about the model. To prevent the divergence of the number constituents of the posterior distribution, the weights of the pairing hypotheses between the multiple detections and the modes of their distribution are evaluated and the mode management technique is applied in implementing the tracking filter. Accordingly, the proposed filter is able to improve computational efficiency while maintaining the estimation performance. Through simulations for typical automotive target tracking scenarios, it is demonstrated that our approach provides reliable target tracking performance compared to the existing filters.

Using dynamic data reconciliation to improve the performance of PID feedback control systems with Gaussian/non-Gaussian distributed disturbance and measurement noise

For a stochastic PID feedback control system, the uncertainty of the working environment often leads to the unsatisfied performance of the system, which does not meet the profit requirements. The working environment generally includes external disturbance and measurement noise, etc. Gaussian distributed measurement noise and disturbances are widely considered while non-Gaussian distributed measurement noise and disturbances are rarely considered. In this paper, the performance degradation of Gaussian/non-Gaussian disturbances and measurement noise on a stochastic PID feedback system is considered and analyzed. An efficient method, dynamic data reconciliation (DDR) is developed to filter measurement noise and disturbances and improve the performance of the stochastic PID feedback control system. By utilizing model-based and measured information, DDR avoids time delays in output estimation. With the detailed theoretical analysis and simulation verification, the effectiveness of the proposed DDR technology on the stochastic PID feedback control system is verified. Compared with conventional exponential filters, DDR can achieve better control performance. The proposed DDR is also used for the control system of the DC–AC​ converter. The improved effect of DDR on the output quality is demonstrated by the results.

Optimal Placement of Phasor Measurement Unit in Smart Grids Considering Multiple Constraints

The distribution of measurement noise is usually assumed to be Gaussian in the optimal phasor measurement unit (PMU) placement (OPP) problem. However, this is not always accurate in practice. This paper proposes a new OPP method for smart grids in which the effects of conventional measurements, limited channels of PMUs, zero-injection buses (ZIBs), single PMU loss contingency, state estimation error (SEE), and the maximum SEE variance (MSEEV) are considered. The SEE and MSEEV are both obtained using a robust <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t$</tex> -distribution maximum likelihood estimator (MLE) because <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t$</tex> -distribution is more flexible for modeling both Gaussian and non-Gaussian noises. The A- and G-optimal experimental criteria are utilized to form the SEE and MSEEV constraints. This allows the optimization problem to be converted into a linear objective function subject to linear matrix inequality observability constraints. The performance of the proposed OPP method is verified by the simulations of the IEEE 14-bus, 30-bus, and 118-bus systems as well as the 188-bus practical distribution system in China.

GSM Filter: A Learning Model of Measurement Noise Distributions for Bayesian Filtering

In this paper, a new universal Bayesian filter with a Gaussian scale mixture (GSM) model, called the Gaussian scale mixture (GSM) filter, for adaptively learning any measurement noise distribution of a state space system is presented. It is demonstrated that the discrete measurement noise distribution of the system takes on a Gaussian sum form when described by a GSM model. It implies that, as the Gaussian sum model does, the GSM model can also be used to approximate any measurement distribution as closely as desired, whatever the probability density function (pdf) of the measurement noise is. If its covariance is known as a priori or can be estimated, it is found that the scale parameter and matrix of the GSM model at each time step can be achieved adaptively to make the following Bayesian inference much simpler. Further analyses show that the GSM model's auxiliary random variable and the system's state posterior at each time step can be inferred by a variational Bayesian (VB) approach adaptively. At each time step, the posterior distribution of system states shown is a Gaussian whatever the measurement noise pdf of the system is. As a result, the Bayesian filtering for nonlinear systems with any measurement noise pdf can be done by our GSM filter, almost as simple as the general Gaussian filters for filtering a nonlinear Gaussian system. Both simulations and experiments are conducted to verify the effectiveness and correctness of the analytical results. The results on the performances of our algorithms fully outperforming those of the existing state of the art Bayesian filters are given.

A Model-Agnostic Method for PMU Data Recovery Using Optimal Singular Value Thresholding

This paper presents a fast model-agnosticmethod for recovering noisy Phasor Measurement Unit (PMU) data streams with missing entries. The measurements are first transformed into a Page matrix, and the original signals are reconstructed using low-rank matrix estimation based on optimal singular value thresholding. Two variations of the recovery algorithm are shown- a) an offline block-processing method for imputing past measurements, and b) an online method for predicting future measurements. Information within a PMU channel (temporal correlation) as well as from different PMU channels in a network (spatial correlation) are utilized to recover degraded data. The proposed method is fast and needs no explicit knowledge of the underlying system model or measurement noise distribution. The performance of the recovery algorithms is illustrated using simulated measurements from the IEEE 39-bus test system as well as real measurements from an anonymized U.S. electric utility. Extensive numeric tests show that the original signals can be accurately recovered in the presence of additive noise, consecutive data drop, as well as simultaneous data erasures across multiple PMU channels.

Design of fractional order PID controller based on minimum variance control and application of dynamic data reconciliation for improving control performance

Fractional order PID (FOPID) has received much attention in recent years and has been applied in different fields. However, the parameter tuning of FOPID is a tough problem. In order to design an optimal FOPID controller, a parameter tuning method based on minimum variance control (MVC) rule is proposed in both SISO and MIMO control systems Considering the influence of measurement noise with Gaussian and non-Gaussian distribution, which is generally ignored in the controller design based on MVC rule, this paper applies dynamic data reconciliation (DDR) technology to suppress its influence and improve the control performance of the designed FOPID. In the case of considering Gaussian and non-Gaussian distributed measurement noise, SISO and MIMO control systems are employed to verify the effectiveness of the proposed FOPID parameter tuning and DDR method. The results show that the designed FOPID improves the performance of the control system, and the DDR technology suppresses the negative effect of measurement noise and improves the control performance of the designed FOPID.

A new Gaussian–Student's t mixing distribution‐based Kalman filter with unknown measurement random delay rate

SummaryIn this article, a new Gaussian–Student's t mixing distribution‐based Kalman filter is presented to investigate the filtering issue for linear stochastic system with unknown measurement random delay rate and non‐stationary heavy‐tailed measurement noise. Firstly, by employing a Bernoulli distributed variable and introducing system state extension method, the form of measurement likelihood function of double measurement noise distributions is converted from the weighted sum to an exponential product. Secondly, the non‐stationary heavy‐tailed measurement noises of current time and last time are modeled as Gaussian–Student's t mixing distributions by introducing extra Bernoulli distributed variables. Thirdly, the variational Bayesian technique is utilized to jointly infer the system state, the Bernoulli distributed variables of measurement noises, distribution mixing probabilities, the intermediate random variables, the Bernoulli distributed variable of measurement delay and the unknown measurement random delay rate. Finally, the effectiveness of the proposed Kalman filter is demonstrated by a target tracking simulation experiment.

A Fast and Robust State Estimator Based on Exponential Function for Power Systems

In realistic power system state estimation, the distribution of measurement noise is usually assumed to be Gaussian while many researcher have verified that it can be non-Gaussian. In this paper, a new robust state estimator based on exponential absolute value function is proposed to address the non-Gaussian measurement noise and outliers. The influence function, a robust statistics tool, is used to obtain the state estimates to reduce its computational burden. A state estimation mean squared error formula of the proposed robust estimator is derived which can be used as a reference in the wide area monitoring system design or upgrade. Simulation results obtained from the IEEE 30-bus, 118-bus and 300-bus systems verify the effectiveness and robustness of the proposed robust estimator.

Automatic differentiation to simultaneously identify nonlinear dynamics and extract noise probability distributions from data

The sparse identification of nonlinear dynamics (SINDy) is a regression framework for the discovery of parsimonious dynamic models and governing equations from time-series data. As with all system identification methods, noisy measurements compromise the accuracy and robustness of the model discovery procedure. In this work we develop a variant of the SINDy algorithm that integrates automatic differentiation and recent time-stepping constrained motivated by Rudy et al (2019 J. Computat. Phys. 396 483–506) for simultaneously (1) denoising the data, (2) learning and parametrizing the noise probability distribution, and (3) identifying the underlying parsimonious dynamical system responsible for generating the time-series data. Thus within an integrated optimization framework, noise can be separated from signal, resulting in an architecture that is approximately twice as robust to noise as state-of-the-art methods, handling as much as 40% noise on a given time-series signal and explicitly parametrizing the noise probability distribution. We demonstrate this approach on several numerical examples, from Lotka-Volterra models to the spatio-temporal Lorenz 96 model. Further, we show the method can learn a diversity of probability distributions for the measurement noise, including Gaussian, uniform, Gamma, and Rayleigh distributions.

Robust adaptive multi‐target tracking with unknown measurement and process noise covariance matrices

A robust adaptive probability hypothesis density (PHD) filter is proposed to address the degradation of PHD performance due to an unknown process noise and measurement noise covariance matrix. Therefore, the inverse Wishart distribution is introduced to model the prior distribution of process noise and measurement noise. Meanwhile, the multi-target posterior intensity is approximated as a mixture of the inverse Wishart distribution and Gaussian distribution. The closed solution of the robust PHD filter is derived by the variational Bayes approach. Simulation results show that the proposed algorithm outperforms the Gaussian-mixed PHD filter and the variational Bayesian PHD filter in terms of target number estimation accuracy and optimal sub-pattern assignment distance.

A Flow Sensor-Based Suction-Index Control Strategy for Rotary Left Ventricular Assist Devices.

Rotary left ventricular assist devices (LVAD) have emerged as a long-term treatment option for patients with advanced heart failure. LVADs need to maintain sufficient physiological perfusion while avoiding left ventricular myocardial damage due to suction at the LVAD inlet. To achieve these objectives, a control algorithm that utilizes a calculated suction index from measured pump flow (SIMPF) is proposed. This algorithm maintained a reference, user-defined SIMPF value, and was evaluated using an in silico model of the human circulatory system coupled to an axial or mixed flow LVAD with 5–10% uniformly distributed measurement noise added to flow sensors. Efficacy of the SIMPF algorithm was compared to a constant pump speed control strategy currently used clinically, and control algorithms proposed in the literature including differential pump speed control, left ventricular end-diastolic pressure control, mean aortic pressure control, and differential pressure control during (1) rest and exercise states; (2) rapid, eight-fold augmentation of pulmonary vascular resistance for (1); and (3) rapid change in physiologic states between rest and exercise. Maintaining SIMPF simultaneously provided sufficient physiological perfusion and avoided ventricular suction. Performance of the SIMPF algorithm was superior to the compared control strategies for both types of LVAD, demonstrating pump independence of the SIMPF algorithm.

A Gaussian mixture regression model based adaptive filter for non-Gaussian noise without a priori statistic

In many engineering systems, the distribution of measurement noise is unknown and non-Gaussian, such as skewed, multimodal, time-varying distributions and we can not obtain these prior statistics in advance. How to achieve state estimation via this kind of non-Gaussian measurement noises without prior statistic information is a challenging problem. In this paper, a novel Gaussian mixture regression model (GMRM) is proposed to model the unknown non-Gaussian measurement likelihood for Bayesian update to achieve nonlinear state estimation. Without any prior assumption or limitation of measurement noises’ statistics and distributions, the GMRM can still describe the measurement likelihood accurately based on a group of parameters which are adjusted by maximizing the evidence lower bound. Based on the optimized GMRM, a new variational Bayesian Gaussian mixture filter is proposed by using the variational Bayesian approach. To eliminate the influence of the initialization of the introduced parameters, a learning scheme is proposed to adaptively optimize their hyper-parameters based on the historical measurements. Finally, simulation examples are employed to illustrate the effectiveness of the filter.

Deep learning-based parameter estimation in fetal diffusion-weighted MRI

Diffusion-weighted magnetic resonance imaging (DW-MRI) of fetal brain is challenged by frequent fetal motion and signal to noise ratio that is much lower than non-fetal imaging. As a result, accurate and robust parameter estimation in fetal DW-MRI remains an open problem. Recently, deep learning techniques have been successfully used for DW-MRI parameter estimation in non-fetal subjects. However, none of those prior works has addressed the fetal brain because obtaining reliable fetal training data is challenging. To address this problem, in this work we propose a novel methodology that utilizes fetal scans as well as scans from prematurely-born infants. High-quality newborn scans are used to estimate accurate maps of the parameter of interest. These parameter maps are then used to generate DW-MRI data that match the measurement scheme and noise distribution that are characteristic of fetal data. In order to demonstrate the effectiveness and reliability of the proposed data generation pipeline, we used the generated data to train a convolutional neural network (CNN) to estimate color fractional anisotropy (CFA). We evaluated the trained CNN on independent sets of fetal data in terms of reconstruction accuracy, precision, and expert assessment of reconstruction quality. Results showed significantly lower reconstruction error (n=100,p<0.001) and higher reconstruction precision (n=20,p<0.001) for the proposed machine learning pipeline compared with standard estimation methods. Expert assessments on 20 fetal test scans showed significantly better overall reconstruction quality (p<0.001) and more accurate reconstruction of 11 regions of interest (p<0.001) with the proposed method.