Prediction Error Covariance Matrix Research Articles

State estimation for linear discrete-time systems with binary-valued quantized innovations against data tampering attacks

In today’s information era, cyber–physical systems (CPS) have been widely used in many industrial and infrastructure fields, and the study of security problems of CPS has become a crucial issue. For linear discrete-time systems, this paper investigates the state estimation problem based on the quantized innovations and transmission data tampering scenarios. From the attacker’s view, by designing the tampering matrix and combining the property of the error function, the optimal attack strategy in the sense of maximizing the prediction error covariance matrix is given. From the defender’s view, the optimal defense scheme based on multiplicative compensation is given in the case of a known attack strategy. Moreover, in the case of unknown attack strategies, a joint estimation algorithm is designed to give the optimal defense scheme subject to error constraints. In this paper, state estimators in recursive form are given for all above cases, and the algorithms are proved to be consistent. The theoretical results obtained are verified by simulations.

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.

A Variational Bayesian Truncated Adaptive Filter for Uncertain Systems with Inequality Constraints

In this paper, a variational Bayesian (VB) truncated adaptive filter for uncertain systems with inequality constraints is proposed. By choosing the skew‐t and inverse Wishart distributions as the prior information of the measurement noise and predicted error covariance matrix, the state vector, the predicted error covariance matrix, and noise parameters are inferred and approximated by using the VB method. To achieve the inequality‐constrained estimation, the constrained state is computed by truncating the probability density function (PDF) of the estimated state after the variational update stage; the mean and covariance of the constrained state are the first and second moments of the truncated PDF. Considering the model uncertainties where the system dynamics are unpredictable, a multiple model VB truncated adaptive filter is proposed in the interacting multiple model framework. The performances of the proposed algorithms are evaluated via the target tracking simulations and the robot positioning experiments. Results show that the proposed algorithms improve estimation accuracy compared with the existing adaptive filters when the states suffer inequality constraints.

Estimation-free Prediction Algorithms

For Time-varying, Time-invariant, and steady-state systems, Kalman Filter can be implemented as a prediction algorithm, since it produces the state prediction and the corresponding prediction error covariance matrix via the state estimation and the corresponding estimation error covariance matrix. Lainiotis Filter is equivalent to Kalman Filter and can be used to compute the prediction. In this paper, for Time-varying, Time-invariant and steady state systems, estimation-free Prediction Algorithms are derived via Kalman and Lainiotis filters; they are equivalent and compute iteratively the prediction and the corresponding prediction error covariance matrix. The estimation and the corresponding estimation error covariance matrix are not needed and are not computed. The proposed estimation-free prediction algorithms are faster than the Kalman filter.

Attitude Determination System for a Cubesat Experiencing Eclipse.

In the context of Kalman filters, the predicted error covariance matrix Pk+1 and measurement noise covariance matrix R are used to represent the uncertainty of state variables and measurement noise, respectively. However, in real-world situations, these matrices may vary with time due to measurement faults. To address this issue in CubeSat attitude estimation, an adaptive extended Kalman filter has been proposed that can dynamically estimate the predicted error covariance matrix and measurement noise covariance matrix using an expectation-maximization approach. Simulation experiments have shown that this algorithm outperforms existing methods in terms of attitude estimation accuracy, particularly in sunlit and shadowed phases of the orbit, with the same filtering parameters and initial conditions.

Maximum correntropy criterion variational Bayesian adaptive Kalman filter based on strong tracking with unknown noise covariances

The performance of the current state estimation will degrade in the existence of slow-varying noise statistics. To solve the aforementioned issues, an improved strong tracking maximum correntropy criterion variational-Bayesian adaptive Kalman filter is presented in this paper. First of all, the inverse-Wishart distribution, as the conjugate-prior, is adopted to model the unknown and time-varying measurement and process noise covariances, then the noise covariances and system state are estimated via the variational Bayesian method. Secondly, the multiple fading-factors are obtained and evaluated to modify the prediction error covariance matrix to address the problems associated with inaccurate error estimation. Finally, the maximum correntropy criterion is employed to correct the filtering gain, which improves the filtering performance of the proposed algorithm. Simulation results show that the proposed filter exhibits better accuracy and convergence performance compared to other existing algorithms.

A novel variational Bayesian adaptive Kalman filter with mismatched process noise covariance matrix

AbstractThis paper proposes a novel variational Bayesian (VB) adaptive Kalman filter with mismatched process noise covariance matrix (PNCM). Firstly, this paper explains the reason why the predicted error covariance matrix (PECM) is chosen for variational inference. Secondly, compared with the earlier VB adaptive Kalman filter (VB‐AKF‐Q), the proposed filter calculate the dynamic model of the PECM with its historical estimation information. Therefore, the proposed filter can overcome the influence of mismatched PNCM on the initial value setting of PECM in the VB‐AKF‐Q. Finally, we use the evidence lower bound for the proposed filter and give the convergence criterion on this basis. Some examples with a target tracking simulation are carried out to demonstrate the superiority of the proposed filter.

Robust interacting multiple model cubature Kalman filter for nonlinear filtering with unknown non-Gaussian noise

Estimating noise covariance matrices and suppressing outliers simultaneously is a huge challenge because they are coupled. In order to solve the nonlinear filtering problem with unknown non-Gaussian noise, the maximum correntropy criterion (MCC) is used to suppress the outliers, while the variational Bayesian (VB) technique is used to estimate the one-step prediction error covariance matrix (PECM) and the measurement noise covariance matrix (MNCM), and the nonlinear problem is solved by the third-order spherical radial cubature rule. In this paper, nominal measurement is constructed to eliminate the influence of outliers on the recursive estimation of MNCM. Through these operations, the variational Bayesian and maximum correntropy based cubature Kalman filter (VBMCCKF) and the maximum correntropy cubature Kalman filter (MCCKF) are derived. The interacting multiple model (IMM) fusion framework is used to promote the collaborative work of VBMCCKF and MCCKF, thus a novel filter is proposed in this paper. The simulation verifies the validity and universal applicability of the proposed filter. The estimation error of the proposed filter in spacecraft autonomous celestial navigation is about 30% less than that of the existing filter with the best performance.

A Lag Compensation-Enhanced Adaptive Quasi-Fading Kalman Filter for Sensorless Control of Synchronous Reluctance Motor

A novel position estimation strategy based on lag compensation-assisted adaptive quasi-fading Kalman filter (LC-AQFKF) is proposed for synchronous reluctance motor (SynRM) sensorless drive in this article. In LC-QFKF, the quasi-fading factor is derived to avoid the harsh assumptions of conventional adaptive fading Kalman filter, and accessibility of the method is improved while ensuring the estimation accuracy. The computational efficiency of LC-AQFKF is promoted by introducing the quasi-fading factor into the prediction error covariance matrix. Moreover, the frequency characteristic of AQFKF-based active back electromotive force observer is analyzed, and the phase lag problem in high-speed situations caused by the low-pass filtering property of AQFKF is overcome. In this way, the estimation accuracy of the rotor position is significantly enhanced. Besides, the double dynamic position compensation method is put forward to strengthen the position estimation performance of sensorless SynRM drive under dynamic conditions. The effectiveness of the proposed scheme is validated at a 1.5 kW SynRM drive.

Strong tracking square-root modified sliding-window variational adaptive Kalman filtering with unknown noise covariance matrices

The Kalman filter’s performance deteriorates in the existence of slowly time-varying and unknown measurement and process noise covariances. A simplified strong tracking square-root modified sliding window variational adaptive Kalman filter is proposed for the aforementioned challenges in this paper. A modified slidingwindow variational adaptive Kalman filtering is designed in the proposed algorithm capable of correcting and smoothing the previous states in accordance with the latter states and reducing the number of backward iterations to improve the filtering accuracy and computational efficiency of the algorithm. The multiple fading factors have been constructed to correct the one-step predicted error covariance matrix. Moreover, the square root decomposition approach is developed for decomposing the error covariance matrices to eliminate numerical rounding errors. The simulation results demonstrate that the proposed algorithm exhibits superior tracking capacity of the one-step predicted error covariance matrix and filtering accuracy compared with the existing filters.

Optimal sensor placement for strain sensing of a beam of high-speed EMU

Aiming at the strain sensing of a beam of high-speed EMU, optimal sensor placement (OSP) is studied based on the information gain and modal expansion method within the Bayesian framework. In addition, the role of the prediction error in OSP and strain reconstruction is investigated in detail. Two new prediction errors are proposed: the measurement error weighted by modal strain energy and the modeling error based on the change of element stiffness. The covariance matrix of prediction error is obtained by probability method, which improves the calculation efficiency. Also, a method to determine the number of modes based on the change rate of modal strain energy is proposed. With reference to a case study of a beam, the effect of the prediction error on OSP and strain sensing is described and the reliability and the robustness of the proposed method are tested. Finally, the proposed method is further validated through a case study of full-scale beam monitoring systems based on Fiber Bragg grating (FBG) sensors. The errors between the real and reconstructed strains are compared through the measured data. The results show that the errors between the reconstructed strain and the real strain are very small, which illustrates that this method can be used for strain sensing.

Theoretical accuracy for indirect predictions based on SNP effects from single-step GBLUP

BackgroundAlthough single-step GBLUP (ssGBLUP) is an animal model, SNP effects can be backsolved from genomic estimated breeding values (GEBV). Predicted SNP effects allow to compute indirect prediction (IP) per individual as the sum of the SNP effects multiplied by its gene content, which is helpful when the number of genotyped animals is large, for genotyped animals not in the official evaluations, and when interim evaluations are needed. Typically, IP are obtained for new batches of genotyped individuals, all of them young and without phenotypes. Individual (theoretical) accuracies for IP are rarely reported, but they are nevertheless of interest. Our first objective was to present equations to compute individual accuracy of IP, based on prediction error covariance (PEC) of SNP effects, and in turn, are obtained from PEC of GEBV in ssGBLUP. The second objective was to test the algorithm for proven and young (APY) in PEC computations. With large datasets, it is impossible to handle the full PEC matrix, thus the third objective was to examine the minimum number of genotyped animals needed in PEC computations to achieve IP accuracies that are equivalent to GEBV accuracies.ResultsCorrelations between GEBV and IP for the validation animals using SNP effects from ssGBLUP evaluations were ≥ 0.99. When all available genotyped animals were used for PEC computations, correlations between GEBV and IP accuracy were ≥ 0.99. In addition, IP accuracies were compatible with GEBV accuracies either with direct inversion of the genomic relationship matrix (G) or using the algorithm for proven and young (APY) to obtain the inverse of G. As the number of genotyped animals included in the PEC computations decreased from around 55,000 to 15,000, correlations were still ≥ 0.96, but IP accuracies were biased downwards.ConclusionsTheoretical accuracy of indirect prediction can be successfully obtained by computing SNP PEC out of GEBV PEC from ssGBLUP equations using direct or APY G inverse. It is possible to reduce the number of genotyped animals in PEC computations, but accuracies may be underestimated. Further research is needed to approximate SNP PEC from ssGBLUP to limit the computational requirements with many genotyped animals.

A novel maximum likelihood and moving weighted average based adaptive Kalman filter

For the state estimation with inaccurate noise statistics, the existing adaptive Kalman filters (AKFs) usually have substantial computational complexity or are not easy to estimate online. Inspired by the fact, a new computationally efficient AKF based on maximum likelihood and moving weighted average (MMAKF) is proposed. Firstly, to reduce computational complexity, instead of estimating the noise covariance matrixes, the maximum likelihood principle is introduced to directly estimate the prediction error covariance matrix and innovation covariance matrix. Subsequently, a new moving weighted average algorithm is designed to optimize the estimated results. Then, a computationally efficient AKF is derived, and its convergence performance and application are discussed. Simulation results for the target tracking example illustrate that the proposed AKF can effectively reduce error caused by inaccurate noise statistics and basically keep simplicity and elegance of the classical KF.

Random sample consensus in decentralized Kalman filter

This paper presents a new decentralized Kalman filter (DKF) framework to detect and isolate faulty nodes in a network of sensors. The Chi-square test is a well-versed fault detection method. However, if error in the predicted state estimate is not within bounds defined by the predicted error covariance matrix, the Chi-square test fails. In order to detect faults (also known as outliers) in measurements, the random sample consensus (RANSAC) algorithm has been widely used in computer vision applications. We propose a novel integration of RANSAC into DKF framework, named as the DKF-RANSAC algorithm, that uses the information and information matrix of the connecting nodes to formulate a hypothesis to detect faulty nodes. An approach for minimizing the iterations in the RANSAC algorithm using the Chi-square test is also presented. The proposed DKF-RANSAC algorithm is validated on a target tracking problem. A simulation study shows that this algorithm identifies the faulty nodes correctly and isolates them, as well as handles incorrect initialization error simultaneously. The proposed DKF-RANSAC algorithm is also compared with the well-known Chi-square test as well as an adaptive DKF.

A novel partitioned matrix‐based parameter update method embedded in variational Bayesian for underwater positioning

In order to meet the requirements of the high-precision positioning for the autonomous underwater vehicles (AUVs) in the complex and time-varying marine environments, a novel partitioned matrix-based parameter update method embedded in variational Bayesian (PMPU-VB) is proposed to deal with the positioning accuracy problem caused by the inaccurate predicted error covariance and measurement noise matrices. By employing the variational Bayesian (VB) method, the accurate predicted error covariance matrix and the measurement noise matrix can be obtained. Subsequently, the PMPU-VB, which employs the accurate predicted error covariance matrix and measurement noise matrix, is used as a substitute of the traditional Gaussian filtering (GF) algorithm, to update the probability density function (PDF) of the state vector. The state vector is defined to follow the Gaussian distribution, and the parameters of the Gaussian distribution are deduced by using the proposed partitioned matrix-based parameter update method. Finally, the accurate position information of the AUV can be obtained. Therefore, the more precise state vector and the error covariance matrix are acquired. The experiments results illustrate that the PMPU-VB has higher estimate accuracy, better stability and robustness than other comparison algorithms.

Partial-Update Kalman Filter for Permanent Magnet Synchronous Motor Estimates Under Intermittent Data

The partial-update Kalman filter (PKF) is an extension of the Schmidt Kalman filter, which can improve the capabilities of the conventional extended Kalman filter for handling model uncertainties and nonlinearities. Herein, we adapt the PKF to estimate the states and parameters of electric machines, particularly in cases with intermittent observations. To account for missing data within the filter, the arrival of new measurements is treated as a Bernoulli process. We show that the estimation error of the proposed filter remains bounded if the system satisfies mild assumptions. Moreover, we show that the prediction error covariance matrix is guaranteed to be bounded if the observation arrival rate has a lower bound. Hardware experiments validate this technique for a surface-mounted permanent magnet synchronous motor.

A novel matrix block algorithm based on cubature transformation fusing variational Bayesian scheme for position estimation applied to MEMS navigation system

It is great significant for autonomous underwater vehicle (AUV) to obtain its position in real time. Great developments in low-cost inertial navigation system (INS) have been made due to outstanding merits in micro-electro-mechanical system (MEMS) technologies. The navigation errors of the MEMS grade inertial measurement unit (IMU) increase greatly over time because of the complex and changeable marine environment. The measurement noise plays an important role in state estimation with high accuracy. However, the accuracy of measurement noise will be degraded due to larger MEMS sensors’ errors. To solve the problem above, a novel algorithm which fuses variational Bayesians into nonlinear filtering is proposed. Firstly, the position information is augmented to the measurement vector. The measurement functions are divided into linear and nonlinear. Secondly, the variational Bayesian (VB) method is used to estimate the probability density function of the state vector, predicted error covariance matrix and measurement noise matrix. In the case of calculating the second order moment estimation, the cubature transformation is used to determine nonlinear integral equations, and the linear integral equations are derived by the Kalman filter. Finally, the accurate state vector and error covariance matrix are obtained. The real underwater experiments are performed and experiment results show that the proposed algorithm has better performance in aspect of positioning accuracy of AUV and robustness than the traditional algorithms.

Adaptive decentralized Kalman filters with non-common states for nonlinear systems

This paper presents two fault tolerance methods for decentralized Kalman filter with non-common states (DKF-NCS) for nonlinear systems. The DKF-NCS is used in a sensor network, where states are not uniform across all nodes. To detect and isolate faulty measurements, the χ2 test has been quite useful, where faulty measurements are detected based on the innovation error and innovation error covariance matrix. However, the innovation error and innovation error covariance matrix are dependent on the predicted state vector and its error covariance along with measurement model. The χ2 distribution test fails, if the predicted state vector is not consistent with its predicted error covariance matrix. Also, due to processing of set of measurements independently, fault detection happens more often in decentralized estimation compared to centralized estimation. Therefore, discarding the valid measurements based on χ2 detector may impact the performance of decentralized estimators significantly. To overcome this problem, we propose two adaptive fault tolerance methods. The first method handles faulty measurements at the assimilation step by applying weighted correction of the information and information matrix. The second method modifies the measurement noise matrix based on a closed-form solution, if fault is detected. These methods are validated using 100 simulation runs for a tracking problem. Overall, the proposed methods are demonstrated to be superior compared to the χ2 test and an existing adaptive extended Kalman filter.

Online Adaptive Kalman Filter for Target Tracking With Unknown Noise Statistics

Considering that the external hostile environment will lead to rapid attenuation of sensor signals, which will make the noise parameters we set different from the actual noise parameters. In this letter, a novel online adaptive Kalman filter (AKF) is investigated with the main focus on inaccurate nonzero mean Gaussian white noise inherent in the filtering model. In the proposed AKF, we employed the expectation maximization algorithm to construct the noise parameter iteration expressions and obtain an approximate solution of the noise parameter. Finally, the derived AKF can effectively estimate the one-step prediction mean vector, the one-step prediction error covariance matrix, and the measurement noise covariance matrix. A classical target tracking simulation results show the effectiveness and stability of the derived AKF.

Multiple fading factors-based strong tracking variational Bayesian adaptive Kalman filter

If the system model or the statistical characteristics of noise are inaccurate, the past measurements will directly affect the accuracy of current state estimation or even lead to filtering divergence. To overcome above difficulties, a multiple fading factors-based strong tracking variational Bayesian adaptive Kalman filter is proposed. Firstly, the inverse Wishart distribution is adopted to model the measurement noise covariance matrix. Secondly, the remodified measurement noise covariance matrix and the innovation covariance matrix estimated by exponential weighting method are employed to construct the scalar fading factor. Next, the multiple fading factors are calculated to correct the predicted error covariance matrix. Finally, the local optimal estimations of measurement noise covariance matrix and state are obtained by variational Bayesian approach. The target tracking simulations verify that the proposed algorithm has better tracking ability for the predicted error covariance matrix and the measurement noise covariance matrix compared with the existing filters.