Estimation Error Covariance Matrix Research Articles

Design of Prediction-Based Estimator for Time-Varying Networks subject to Communication Delays and Missing Data

This paper is concerned with the robust optimal estimation problem based on the prediction compensation mechanism for dynamical networks with time-varying parameters, where communication delays and degraded measurements are considered. The missing measurements are characterized by some random variables governed by Bernoulli distribution, where each sensor having individual missing probability is refected. During the signal transmissions through the communication networks, the network-induced communication delays commonly exist among the adjacent nodes transmissions and a prediction updating method is given to compensate the caused impacts. Accordingly, a time-varying state estimator with hybrid compensation scheme is constructed such that, for both the communication delays and missing measurements, a minimized upper bound matrix with regards to the estimation error covariance matrix is found and an explicit estimator parameter matrix is designed at each sampling step accordingly. Finally, the comparative simulations are given to validate the advantages of main results.

A switched SDRE filter for state of charge estimation of lithium-ion batteries

Lithium-ion (Li-ion) batteries need very precise monitor of the state of charge (SOC) to ensure a long cycle life. Hence, a knowledge of the SOC is important for Li-ion batteries. Although SOC cannot be measured directly, it can be estimated from direct measurement variables based on a model of the battery. Single-Particle-Model (SPM), a reduced-order nonlinear electrochemical model, is commonly used for this purpose. State-dependent-Riccati-equation (SDRE) filter is chosen as the estimator due to its high-flexibility in handling the model’s nonlinearity. However, performance of this filter is limited in presence of uncertainties. To tackle this problem, in this paper, a switching concept is induced into SDRE filter, in the form of switched estimation error covariance matrix with a certain frequency. Thus, by changing the Riccati equation dynamic in SDRE filter and proper adjustment of estimation error covariance matrix eigenvalues, performance and robustness of the common SDRE filter is significantly improved for Li-ion SOC estimation. To analyze the fidelity of such a filter in further applications, stability analysis is carried out on a class of nonlinear systems, and ultimate bound of estimation error is analytically obtained, and the influence of switching is investigated. Simulation results reveal effectiveness of the proposed filter compared to common SDRE filter, extended Kalman filter and variable structure approaches. Furthermore, experimental results verify the effectiveness of the proposed method in practice.

Secure dimensionality reduction fusion estimation against eavesdroppers in cyber–physical systems

This paper studies the distributed dimensionality reduction fusion estimation problem for cyber-physical systems with limited bandwidth in presence of eavesdroppers. Since wireless communication is implemented by broadcasting, the eavesdroppers can collude to collect the data through anther communication networks. To protect data privacy, based on the physical processes and local estimation error covariance (EEC) matrix, an insertion method of artificial noise (AN) is developed such that only eavesdroppers’ fusion EEC becomes worse. Meanwhile, the fusion center needs to decode the received signal due to the noise interference, while the successful decoding probability varies with signal to noise ratio. Subsequently, some criteria for the selection probabilities and the successful decoding probabilities are given to guarantee the effectiveness of the AN insertion strategy. Moreover, a sufficient condition of the designed AN power is derived to guarantee the confidentiality. Simulation examples are given to show the effectiveness of the proposed methods.

Nonlinear Filtering Method Using a Switching Error Model for Outlier-Contaminated Observations

This article considers a nonlinear filtering method for handling outliers. The presence of outliers that are gross observation errors can greatly reduce the accuracy of filtering methods that assume Gaussian distributed errors. There are some existing methods that assume a Gaussian observation error, and estimate the error covariance matrix at each time step to avoid overfitting to the outliers. However, the estimates of the covariance matrix under such methods can become unstable. This results in underfitting or overfitting to observations when filtering. This paper presents a new method for handling outliers in nonlinear filtering problems by extending the unscented Kalman filter. In this method, two Gaussian observation error models with distinct covariance matrices are used: one for observations with regular errors, and another with a larger variance specified by a scale parameter for outliers. In addition to the system state, this method estimates an indicator variable that switches between the two models and the scale parameter for outliers. With the inclusion of the indicator variable and scale parameter, the estimated error covariance matrix can handle both regular observations and outliers appropriately at each time step. Furthermore, by estimating the scale parameter, the proposed method can be applied to a dataset without additional tuning to account for outlier characteristics. Through numerical experiments, we find that our method can estimate the system state better than existing methods for both datasets with and without outliers.

Robust Forecasting-Aided State Estimation for Power System Against Uncertainties

Accurate forecasting-aided state estimation plays a vital role in reliable and secure operation of power systems. However, most of existing methods are unable to deal with the uncertainties that might be caused by uncertain model parameters or uncertain noise statistics. Therefore, the performance of these methods may be inevitably degraded significantly. To address these issues, based on the robust control theory, in this paper, by incorporating the modified innovation based Sage-Husa estimator of noise statistics and the proposed estimation error covariance matrix adaptive technique, a novel adaptive $H_{\infty }$ extended Kalman filter (AHEKF) is developed to realize robust forecasting-aided state estimation for power system with model uncertainties. Extensive simulations carried out on several different test systems demonstrate the efficiency and robustness of the proposed method.

Multidimensional Gains for Stochastic Approximation.

This paper deals with iterative Jacobian-based recursion technique for the root-finding problem of the vector-valued function, whose evaluations are contaminated by noise. Instead of a scalar step size, we use an iterate-dependent matrix gain to effectively weigh the different elements associated with the noisy observations. The analytical development of the matrix gain is built on an iterative-dependent linear function interfered by additive zero-mean white noise, where the dimension of the function is M ≥ 1 and the dimension of the unknown variable is N ≥ 1 . Necessary and sufficient conditions for M ≥ N algorithms are presented pertaining to algorithm stability and convergence of the estimate error covariance matrix. Two algorithms are proposed: one for the case where M ≥ N and the second one for the antithesis. The two algorithms assume full knowledge of the Jacobian. The recursive algorithms are proposed for generating the optimal iterative-dependent matrix gain. The proposed algorithms here aim for per-iteration minimization of the mean square estimate error. We show that the proposed algorithm satisfies the presented conditions for stability and convergence of the covariance. In addition, the convergence rate of the estimation error covariance is shown to be inversely proportional to the number of iterations. For the antithesis , contraction of the error covariance is guaranteed. This underdetermined system of equations can be helpful in training neural networks. Numerical examples are presented to illustrate the performance capabilities of the proposed multidimensional gain while considering nonlinear functions.

State estimation for bilinear systems through minimizing the covariance matrix of the state estimation errors

SummaryThis paper considers the state estimation problem of bilinear systems in the presence of disturbances. The standard Kalman filter is recognized as the best state estimator for linear systems, but it is not applicable for bilinear systems. It is well known that the extended Kalman filter (EKF) is proposed based on the Taylor expansion to linearize the nonlinear model. In this paper, we show that the EKF method is not suitable for bilinear systems because the linearization method for bilinear systems cannot describe the behavior of the considered system. Therefore, this paper proposes a state filtering method for the single‐input–single‐output bilinear systems by minimizing the covariance matrix of the state estimation errors. Moreover, the state estimation algorithm is extended to multiple‐input–multiple‐output bilinear systems. The performance analysis indicates that the state estimates can track the true states. Finally, the numerical examples illustrate the specific performance of the proposed method.

Robust dynamic state estimation of power systems with model uncertainties based on adaptive unscented filter

This study considers the dynamic state estimation of power systems with model uncertainties that might be caused by the unknown noise statistics or unpredicted changes to the model parameters. To deal with these issues, an innovation-based estimator that is able to dynamically revise the statistics of system and measurement noise is proposed firstly. Then, based on the H ∞ criteria for bounding the adverse influences on the estimation error of model uncertainties and unscented transform technique, an adaptive strategy is developed to adjust the estimation error covariance matrix under various conditions. Finally, by incorporating the proposed approaches and H ∞ filter theory, a novel adaptive unscented H ∞ filter is established to realise dynamic state estimation of power system against model uncertainties. Extensive simulation results obtained from the IEEE-39 bus test system are presented to illustrate the effectiveness and robustness of the proposed method.

Analytical and Iterative Solutions to GNSS Attitude Determination Problem in Measurement Domain

Attitude determination using double-differenced GNSS carrier phase measurements is studied. A realistic stochastic model is employed to take the correlations among the double-differenced measurements into full consideration. Two important issues concerning iteratively solving the nonlinear least-squares attitude determination problem are treated, namely, the initial guess and the iteration scheme. An analytical and sub-optimal solution is employed to provide the initial guess. In this solution, the orthogonal and determinant constraints among the elements of the direction cosine matrix (DCM) of the attitude are firstly ignored, and hence a relaxed 3×3 matrix is estimated using the linear weighted least-squares method. Then a mathematically feasible DCM, i.e., orthogonal and with +1 determinant, is extracted from the relaxed matrix estimate, optimally in the sense of minimum Frobenius norm. This analytical initial guess estimation method can be used for all feasible cases, including some generated ones, e.g., the case with only 3 antennas and only 3 satellites, subject possibly to some necessary, yet minor modifications. In each iteration, an error attitude, whose DCM is parameterized using the Gibbs vector, is introduced to relate the previously estimated and the true DCM. By linearizing the measurement model at the zero Gibbs vector, the least-squares estimate of the Gibbs vector is obtained and then used to correct the previously estimated DCM. By repeating this process, the truly least-squares estimate of the attitude can be achieved progressively. These are in fact Gauss-Newton iterations. For the final estimate, the variance covariance matrix (VCM) of the attitude estimation error can be retained to evaluate or predict the estimation accuracy. The extraction of the widely used roll-pitch-yaw angles and the VCM of their additive estimation errors from the final solution is also presented. Numerical experiments are conducted to check the performance of the developed theory. For the case with 3 2-meter long and orthogonally mounted baselines, 5 visible satellites, and 5-millimeter standard deviations of the carrier phase measurements, the root mean squared errors (RMSE) of the roll-pitch-yaw angles in the analytical solution are well below 0.5 degrees, and the estimates converge after only one iteration, with all three RMSEs below 0.2 degrees.

Adaptive Robust Cubature Kalman Filter for Power System Dynamic State Estimation Against Outliers

This paper develops an adaptive robust cubature Kalman filter (ARCKF) that is able to mitigate the adverse effects of the innovation and observation outliers while filtering out the system and measurement noises. To develop the ARCKF dynamic state estimator, a batch-mode regression form in the framework of cubature Kalman filter is first established by processing the predicted state and measurement data information simultaneously. Subsequently, based on the regression form, the outliers can be detected and downweighted by the robust projection statistics approach. Then, the adverse effects of innovation and observation outliers can be effectively suppressed by the generalized maximum likelihood (GM)-type estimator utilizing the iteratively reweighted least squares approach. Finally, an adaptive strategy is developed to adjust the state estimation error covariance matrix under different conditions. Extensive simulation results obtained from the IEEE New England 10-machine 39-bus test system under various operating conditions demonstrate the effectiveness and robustness of the proposed method, which is able to track the transients of power system in a more reliable way than the conventional cubature Kalman filter (CKF) and the unscented Kalman filter (UKF).

Robust Measurement Placement for Distribution System State Estimation

This paper considers the problem of placing a limited number of measurements to improve the estimation accuracy in distribution system state estimation (DSSE). In distribution systems, intermittent distributed energy resources and volatile loads will result in a wide variation of system operating conditions. The proposed measurement placement problem is to decide the optimal locations and types of measurements to be placed in the distribution systems that minimize the worst-case estimation errors for DSSE over different system operating conditions. Four indices of the estimation error covariance matrix are chosen as the criteria of accuracy. The proposed measurement placement problem is formulated as a mixed-integer semidefinite programming (MISDP) problem. To avoid the combinatorial complexity, a convex relaxation, followed by a local optimization method, is employed to solve the MISDP problem. The proposed problem and the effectiveness of the proposed solution method are numerically demonstrated on the 33-bus distribution system.

A Sparse-Based Off-Grid DOA Estimation Method for Coprime Arrays

Recently, many sparse-based direction-of-arrival (DOA) estimation methods for coprime arrays have become popular for their excellent detection performance. However, these methods often suffer from grid mismatch problem due to the discretization of the potential angle space, which will cause DOA estimation performance degradation when the target is off-grid. To this end, we proposed a sparse-based off-grid DOA estimation method for coprime arrays in this paper, which includes two parts: coarse estimation process and fine estimation process. In the coarse estimation process, the grid points closest to the true DOAs, named coarse DOAs, are derived by solving an optimization problem, which is constructed according to the statistical property of the vectorized covariance matrix estimation error. Meanwhile, we eliminate the unknown noise variance effectively through a linear transformation. Due to finite snapshots effect, some undesirable correlation terms between signal and noise vectors exist in the sample covariance matrix. In the fine estimation process, we therefore remove the undesirable correlation terms from the sample covariance matrix first, and then utilize a two-step iterative method to update the grid biases. Combining the coarse DOAs with the grid biases, the final DOAs can be obtained. In the end, simulation results verify the effectiveness of the proposed method.

Fusion estimation in clustering sensor networks under stochastic deception attacks

ABSTRACTIn this paper, a fusion estimation problem in clustering sensor networks under stochastic deception attacks is investigated. The sensors are divided into several clusters, and a local estimator is embedded in each cluster. During the network transmissions, deception attacks are considered to randomly occur and the statuses of the attacks are described by a set of Bernoulli variables. By using the clustering sampled information, a local minimum variance filter is derived and the filter gains are recursively determined. By using the local estimates and the local estimation error covariance matrices, a covariance intersection fusion algorithm is presented for the fusion centre to generate the final estimate. Moreover, the relationship between the trace of the local estimation error covariance matrix and the uniform successful attack rate is analysed. An illustrative example is given to shown the effectiveness of the proposed theoretical results.

Subarray Processing for Projection-based RFI Mitigation in Radio Astronomical Interferometers

Abstract Radio Frequency Interference (RFI) is a major problem for observations in Radio Astronomy (RA). Adaptive spatial filtering techniques such as subspace projection are promising candidates for RFI mitigation; however, for radio interferometric imaging arrays, these have primarily been used in engineering demonstration experiments rather than mainstream scientific observations. This paper considers one reason that adoption of such algorithms is limited: RFI decorrelates across the interferometric array because of long baseline lengths. This occurs when the relative RFI time delay along a baseline is large compared to the frequency channel inverse bandwidth used in the processing chain. Maximum achievable excision of the RFI is limited by covariance matrix estimation error when identifying interference subspace parameters, and decorrelation of the RFI introduces errors that corrupt the subspace estimate, rendering subspace projection ineffective over the entire array. In this work, we present an algorithm that overcomes this challenge of decorrelation by applying subspace projection via subarray processing (SP-SAP). Each subarray is designed to have a set of elements with high mutual correlation in the interferer for better estimation of subspace parameters. In an RFI simulation scenario for the proposed ngVLA interferometric imaging array with 15 kHz channel bandwidth for correlator processing, we show that compared to the former approach of applying subspace projection on the full array, SP-SAP improves mitigation of the RFI on the order of 9 dB. An example of improved image synthesis and reduced RFI artifacts for a simulated image “phantom” using the SP-SAP algorithm is presented.

Decentralized state estimation for a large-scale spatially interconnected system

A decentralized state estimator is derived for the spatially interconnected systems composed of many subsystems with arbitrary connection relations. An optimization problem on the basis of linear matrix inequality (LMI) is constructed for the computations of improved subsystem parameter matrices. Several computationally effective approaches are derived which efficiently utilize the block-diagonal characteristic of system parameter matrices and the sparseness of subsystem connection matrix. Moreover, this decentralized state estimator is proved to converge to a stable system and obtain a bounded covariance matrix of estimation errors under certain conditions. Numerical simulations show that the obtained decentralized state estimator is attractive in the synthesis of a large-scale networked system.

Statistically robust transceiver design algorithms for relay aided multiple‐input multiple‐output interference systems

In this study, the authors consider robust transceiver design for amplify-and-forward relay aided multiple-input multiple-output interference systems assuming direct transmitter–receiver links and imperfect channel state information (CSI). The imperfect CSI of each link consists of the estimated channel and the covariance matrix of the channel estimation error. The authors address two transceiver optimisation problems: one is minimising the sum of averaged mean squared error (MSE) at all the receivers and the other is minimising the maximum averaged MSE among all the receivers, both of which are subject to power constraints at the transmitter and relay nodes. The formulated sum-MSE minimisation and max-MSE minimisation based optimisation problems are non-convex with matrix variables and therefore a globally optimal solution is difficult to obtain. To solve these non-convex optimisation problems, they develop sub-optimal iterative algorithms based on alternating minimisation approach to jointly optimise the precoding matrices at the transmitter and relay nodes and the receiver filter matrices. Simulation results demonstrate the effectiveness of the proposed algorithms and their improved performance against CSI uncertainties at similar computational cost as the existing non-robust designs.

Event-Based Sensing and Control for Remote Robot Guidance: An Experimental Case.

This paper describes the theoretical and practical foundations for remote control of a mobile robot for nonlinear trajectory tracking using an external localisation sensor. It constitutes a classical networked control system, whereby event-based techniques for both control and state estimation contribute to efficient use of communications and reduce sensor activity. Measurement requests are dictated by an event-based state estimator by setting an upper bound to the estimation error covariance matrix. The rest of the time, state prediction is carried out with the Unscented transformation. This prediction method makes it possible to select the appropriate instants at which to perform actuations on the robot so that guidance performance does not degrade below a certain threshold. Ultimately, we obtained a combined event-based control and estimation solution that drastically reduces communication accesses. The magnitude of this reduction is set according to the tracking error margin of a P3-DX robot following a nonlinear trajectory, remotely controlled with a mini PC and whose pose is detected by a camera sensor.

A quadratic-cost dual control-based approach for optimal trajectory planning under uncertainty

This paper presents a dual control-based approach for optimal trajectory planning under uncertainty. The method approximately converts a nonlinear stochastic optimal control problem whose objective function is a combination of quadratic stage and/or terminal costs, with additive Gaussian process and measurement noises, into a deterministic optimal control problem by augmenting the uncertainty state defined by the square-root of the estimation error covariance matrix. The open-loop solution to the resulting deterministic optimal control reformulation is obtained using an existing pseudo-spectral method. The effectiveness of the proposed dual control-based approach is verified with two numerical examples of trajectory planning for two-dimensional robot motion with lack of observability for localization, which highlights the impact of the dual effect on the shape of designed paths.

A Robust Iterated Extended Kalman Filter for Power System Dynamic State Estimation

This paper develops a robust iterated extended Kalman filter (EKF) based on the generalized maximum likelihood approach (termed GM-IEKF) for estimating power system state dynamics when subjected to disturbances. The proposed GM-IEKF dynamic state estimator is able to track system transients in a faster and more reliable way than the conventional EKF and the unscented Kalman filter (UKF) thanks to its batch-mode regression form and its robustness to innovation and observation outliers, even in position of leverage. Innovation outliers may be caused by impulsive noise in the dynamic state model while observation outliers may be due to large biases, cyber attacks, or temporary loss of communication links of PMUs. Good robustness and high statistical efficiency under Gaussian noise are achieved via the minimization of the Huber convex cost function of the standardized residuals. The latter is weighted via a function of robust distances of the two-time sequence of the predicted state and innovation vectors and calculated by means of the projection statistics. The state estimation error covariance matrix is derived using the total influence function, resulting in a robust state prediction in the next time step. Simulation results carried out on the IEEE 39-bus test system demonstrate the good performance of the GM-IEKF under Gaussian and non-Gaussian process and observation noise.

Power scheduling for Kalman filtering over lossy wireless sensor networks

With the goal of monitoring physical processes, a wireless sensor network (WSN) is often deployed along with a fusion center to estimate the state of general linear stochastic systems. As WSNs comprise a large number of low-cost, battery-driven sensor nodes with limited transmission bandwidth, conservation of transmission resources (power and bandwidth) is of paramount importance. In this context, the present study considers power scheduling for Kalman filtering (KF) using scalar messages exchanged over wireless sensor links, where random measurement packet drops are possible. Each sensor node sequentially decides whether a high or low transmission power is needed to communicate its scalar observations based on a rule that promotes power scheduling with minimal impact on the state estimator's mean-squared error. Assuming approximately Gaussian state predictors, the minimum mean-squared error optimal power schedule is developed for KF that also accounts for dropped data packets. Leveraging statistical convergence characteristics of the estimation error covariance matrix, both sufficient and necessary conditions are established that guarantee the stability of the resultant KF estimator.