Predicted State Vector Research Articles

Innovative Formulation in Discrete Kalman Filtering with Constraints - A Generic Framework for Comprehensive Error Analysis

This manuscript establishes a generic framework for comprehensive error analysis in discrete Kalman filtering with constraints, which systematically provides a complete set of algorithmic formulas along with demonstrating an alternative process of theoretical analytics of discrete Kalman filter. This constructive work aims extensively to standardize the formulation of Kalman filter with constraints. In analogy to the similar framework for standard discrete Kalman filter (without any constraints), the proposed framework specifically considers: model formulation vs. the error sources, the solution of the state and process noise vectors, the residuals for the process noise vector and the measurement noise vector, the redundancy contribution of the predicted state vector, process noise vector and measurement vector, and other relevant essential aspects, of which some of the features are essential to comprehensive error analysis, but are nonexistent yet in the primary algorithm in Kalman filtering with constraints. Besides, the algorithmic form of the Extended Kalman filter with constraints is also provided for practical purpose. At the end, specific remarks about the developed framework are given to emphasize on its usage to a certain extend.

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.

AGMC-Based Robust Cubature Kalman Filter for SINS/GNSS Integrated Navigation System With Unknown Noise Statistics

A new robust cubature Kalman filter is proposed using adaptive generalized maximum correntropy (AGMC) criterion rather than the conventional MMSE criterion in this paper. In the proposed method, the adaptive generalized maximum correntropy (AGMC) criterion is firstly constructed from an adaptive forgetting correntropy based cost function, which is rather robust with respect to the process uncertainty and non-Gaussian noise. On this basis, a new robust cubature Kalman filter is further derived, where the predicted state vector and received measurements are processed simultaneously based on the regression form derived via the statistical linearization approach. An adaptive forgetting scheme is then proposed in combination with the AGMC-CKF to update the parameters of the AGMC adaptively in real time. Taking advantage of the AGMC, the unknown noise statistics caused by the process uncertainty and non-Gaussian noise can be effectively suppressed. Simulations and car-mounted experiments demonstrate that the proposed filter is superior in terms of estimation accuracy and robustness as compared with the related state-of-art methods.

A Novel Adaptive Filter for Cooperative Localization Under Time-Varying Delay and Non-Gaussian Noise

In this study, to cope with time-varying delay in acoustic communication under non-Gaussian noise caused by outliers, a novel robust delay filtering algorithm is proposed for cooperative localization of autonomous underwater vehicles (AUVs). First, the modified measurement equations of the nonlinear cooperative location system with time-varying delay are derived, and the delay caused by the information processing and propagation of the underwater acoustic modem is converted into measurement bias. Second, to improve the robustness of the system with outliers caused by abnormal measurement values of underwater acoustic communication and Doppler velocity log (DVL), the statistical similarity measure (SSM) is introduced to construct the cost function. This is accomplished by quantifying the similarity between the state vector and the predicted state vector, similarity between the delay measurement, and modified predicted delay measurement. Finally, the modified measurement noise variance is obtained by maximizing the cost function, and two algorithms for the approximate solution of the nonlinear cost function are presented. The proposed robust delayed algorithm alleviates the impacts of delayed measurement with outliers on localization accuracy. The effectiveness and potential of the proposed filter are verified by lake trials.

Robust Unscented Kalman Filter for Power System Dynamic State Estimation With Unknown Noise Statistics

Due to the communication channel noises, GPS synchronization process, changing environment temperature and different operating conditions of the system, the statistics of the system process and measurement noises may be unknown and they may not follow Gaussian distributions. As a result, the traditional Kalman filter-based dynamic state estimators may provide strongly biased state estimates. To address these issues, this paper develops a robust generalized maximum-likelihood unscented Kalman filter (GM-UKF). The statistical linearization approach is presented to derive a compact batch-mode regression form by processing the predicted state vector and the received measurements simultaneously. This regression form enhances the data redundancy and allows us to detect bad phasor measurement unit measurements and incorrect state predictions, and filter out unknown Gaussian and non-Gaussian noises through the generalized maximum likelihood-estimator. The latter minimizes a convex Huber function with weights calculated via the projection statistics (PS). Particularly, the PS is applied to a proposed 2-dimensional matrix that consists of temporally correlated innovation vectors and predicted states. Finally, the total influence function is used to derive the error covariance matrix of the GM-UKF state estimates, yielding the robust state prediction at the next time instant. Extensive simulations carried out on the IEEE 39-bus test system demonstrate the effectiveness and robustness of the proposed method.

Adaptive Square-Root Unscented Particle Filtering Algorithm for Dynamic Navigation.

This paper presents a new adaptive square-root unscented particle filtering algorithm by combining the adaptive filtering and square-root filtering into the unscented particle filter to inhibit the disturbance of kinematic model noise and the instability of filtering data in the process of nonlinear filtering. To prevent particles from degeneracy, the proposed algorithm adaptively adjusts the adaptive factor, which is constructed from predicted residuals, to refrain from the disturbance of abnormal observation and the kinematic model noise. Cholesky factorization is also applied to suppress the negative definiteness of the covariance matrices of the predicted state vector and observation vector. Experiments and comparison analysis were conducted to comprehensively evaluate the performance of the proposed algorithm. The results demonstrate that the proposed algorithm exhibits a strong overall performance for integrated navigation systems.

Robust Forecasting Aided Power System State Estimation Considering State Correlations

With the increase of load fluctuations and the integration of stochastic distributed generations (DGs), there have been more and more research interests in forecasting-aided state estimation. In this paper, we propose a robust generalized maximum likelihood (GM)-estimator based power system forecasting-aided state estimation, which integrates the statistical characteristics of both loads and DGs, i.e., spatial and temporal correlations. A first order vector auto-regressive model (VAR(1)) is developed to capture the statistical characteristics of load and DGs, facilitating short-term loads and DGs forecasting. These forecasted power injections are further combined with power balance equations to derive a new state transition model, where the relationship between forecasted state vector and predicted power injections is expressed explicitly. After that, a redundant batch regression model that simultaneously processes predicted state vector and received observations is derived, allowing the development of a robust estimator. To this end, we propose a robust GM-estimator that leverages modified projection statistics and a Huber convex score function, to bound the influence of observation outliers while maintaining its high statistical estimation efficiency. Finally, the iteratively reweighted least squares algorithm is adopted to solve the GM-estimator. Numerical comparisons on IEEE benchmark systems with DGs integration demonstrate the efficiency and robustness of the proposed method.

AMA- and RWE- Based Adaptive Kalman Filter for Denoising Fiber Optic Gyroscope Drift Signal

An improved double-factor adaptive Kalman filter called AMA-RWE-DFAKF is proposed to denoise fiber optic gyroscope (FOG) drift signal in both static and dynamic conditions. The first factor is Kalman gain updated by random weighting estimation (RWE) of the covariance matrix of innovation sequence at any time to ensure the lowest noise level of output, but the inertia of KF response increases in dynamic condition. To decrease the inertia, the second factor is the covariance matrix of predicted state vector adjusted by RWE only when discontinuities are detected by adaptive moving average (AMA).The AMA-RWE-DFAKF is applied for denoising FOG static and dynamic signals, its performance is compared with conventional KF (CKF), RWE-based adaptive KF with gain correction (RWE-AKFG), AMA- and RWE- based dual mode adaptive KF (AMA-RWE-DMAKF). Results of Allan variance on static signal and root mean square error (RMSE) on dynamic signal show that this proposed algorithm outperforms all the considered methods in denoising FOG signal.

Random weighting estimation of kinematic model error for dynamic navigation

This paper presents a new random weighting method to deal with the systematic error of the kinematic model for dynamic navigation. This method incorporates random weights in the kinematic model to control the systematic error of the kinematic model for improving the navigation accuracy. A theory of random weighting estimation is established, showing that 1) the random weighting estimation of the kinematic model’s systematic error is unbiased and 2) the covariance matrix of the predicted state vector can be controlled by adjusting the covariance matrices of the predicted residual vector and estimated state vector to improve the accuracy of state prediction. Random weighting estimations are also constructed for the systematic error of the kinematic model as well as the covariance matrices of predicted residual vector, predicted state vector, and state noise vector. Experimental results demonstrate the effectiveness of the proposed random weighting method in resisting the disturbances of the kinematic model noise for improving the accuracy of dynamic navigation.

Sage windowing and random weighting adaptive filtering method for kinematic model error

This paper presents a new method for adaptive estimation of kinematic model error in dynamic aircraft navigation. This method combines the concepts of random weighting and Sage windowing to online monitor predicted and observation residuals to control the influence of the kinematic model's systematic error on system state estimation. Based on the Sage windowing, random weighting estimations are constructed within a moving time window for the systematic error of the kinematic model as well as the covariance matrices of the observation noise vector, the predicted residual vector, and the predicted state vector. Experimental results and comparison analysis demonstrate that the proposed method not only adjusts the covariance matrices of the observation noise vector and the predicted residual vector, but also effectively controls the influence of the kinematic model error on state parameter estimation, thus improving the navigation accuracy.

Windowing-based random weighting fitting of systematic model errors for dynamic vehicle navigation

The Kalman filter is a commonly used computational method for dynamic vehicle navigation and positioning. However, it requires kinematic and observation models not contain any systematic error; otherwise, the resultant navigation solution will be biased or even divergent. In order to overcome this limitation, this paper presents a new windowing-based random weighting method to fit the systematic errors of kinematic and observation models within a moving time window for dynamic vehicle navigation. This method compensates the systematic model errors by correcting observation residual vector and state noise vector during the filtering process. Random weighting theories are established to fit the systematic model errors and the covariance matrices of observation vector and predicted state vector within a moving time window. Experiments and comparison analysis with the existing methods demonstrate that the proposed method can effectively resist the disturbances on system state estimation due to the systematic errors of kinematic and observation models, thus significantly improving the accuracy of dynamic vehicle navigation.

A Fuzzy Robust Filtering Method for Transfer Alignment

This paper presents a fuzzy anti-interference transfer alignment method for airborne strapdown inertial navigation system (SINS). This fuzzy anti-interference transfer alignment method takes the influence of systematic error into account for SINS transfer alignment. The fuzzy rules are constructed and incorporated in the filtering process to estimate the covariance matrices of observation vector and predicted state vector with random weight method. The experimental results demonstrate that the proposed method can resist the interferences caused by the airborne maneuvering process, thus improving the accuracy for SINS transfer alignment.

Nonlinear model-predictive control with hysteresis compensation of an electro-pneumatic clutch for truck applications

This contribution presents two real-time capable nonlinear model-predictive control (NMPC) approaches for an electro-pneumatic clutch for heavy trucks: a centralized control approach and a cascaded one. A clutch is necessary at start-up or during gear shifts to connect or disconnect the combustion engine and the gear box. This automated actuator disburdens the driver and provides the necessary actuation force according to the large torque typically transmitted through the drive train. The force characteristic of the clutch, however, is subject to hysteresis, which is described by a generalized Bouc–Wen model and used for a feedforward hysteresis compensation in the control algorithm. The proposed NMPC-algorithm involves (i) a minimization of the difference between the desired and predicted state vector at the end of the prediction horizon and (ii) flatness-based techniques to compute desired trajectories for the complete state vector as well as the control input. The optimal control is given by an additional, minimum-norm control input that minimizes the difference between the predicted state vector and the desired one at the end of the prediction horizon. Thereby, the computation effort of the NMPC approaches can be kept relatively small, and a real-time evaluation becomes possible. A reduced-order observer estimates an effective pressure in the clutch that also accounts for an uncertain disturbance force. Thereby, a disturbance compensation and a high tracking accuracy is achievable for the piston position as controlled variable. The efficiency of the two proposed control structures is emphasized by experimental results from a dedicated test rig.

Statistical properties of the state obtained by solving a nonlinear multivariate inverse problem

This article takes a statistical approach to solving a multivariate state‐space problem where many data are nonlinearly related to a state vector. The state is unknown and to be predicted, but the problem can be ill posed. A state‐space model quantifies the variability of the physical process (state equation) and of the measurements related to the process (measurement equation). The resulting posterior distribution is then maximized, yielding the predicted state vector. Statistical properties of the predicted state vector, in particular its first two moments with respect to the joint distribution, are approximated using the delta method. These are then applied to the problem of retrieving, from satellite data, a profile of CO2 values in a column of the atmosphere. Copyright © 2012 John Wiley & Sons, Ltd.

Stochastic Interpolation of Spatial Random Fields by BF/MCF-ISM

In the past, interpolation of random fields was successfully treated by Kriging methods for Gaussian fields, and by conditional simulation techniques for a class of non-Gaussian translation fields. Recently, bootstrap filter/Monte Carlo filter (BF/MCF) is extensively used for interpolation of general non-Gaussian fields. However, while BF/MCF is a versatile tool to interpolate non-Gaussian fields, that is an algorithm of generating a set of sample realizations of both a predicted state vector and a filtered state vector, the computational cost is expensive due to the required sample size. In order to reduce the required sample size, an importance sampling function derived from the updating theory of Gaussian fields is applied to the ordinary BF/MCF. Interpolation of spatial fields is first demonstrated by using numerically simulated data, and the BF/MCF incorporated with importance sampling technique (BF/MCF-ISM) for the state estimation of conditional non-Gaussian fields is performed with respect to its efficiency in variance reduction.

An Optimal Adaptive Kalman Filter

In a robustly adaptive Kalman filter, the key problem is to construct an adaptive factor to balance the contributions of the kinematic model information and the measurements on the state vector estimates, and the corresponding learning statistic for identifying the kinematic model biases. What we pursue in this paper are some optimal adaptive factors under the particular conditions that the state vector can or cannot be estimated by measurements. Two optimal adaptive factors are derived, one of which is deduced by requiring that the estimated covariance matrix of the predicted residual vector equals the corresponding theoretical one. The other is obtained by requiring that the estimated covariance matrix of the predicted state vector equals its theoretical one. The two related optimal adaptive factors are given. These are analyzed and compared in theory and in an actual example. This shows, through the actual computations, that the filtering results obtained by optimal adaptive factors are superior to those obtained by adaptive factors based on experience.

State interpolation for on-board navigation systems

Common concepts for autonomous on-board navigation systems rely on the numerical integration of a spacecraft trajectory between subsequent measurements of a navigation sensor such as GPS. In combination with a Kalman filter, a predicted state vector becomes available at discrete, but not necessarily equidistant time steps. When used for real-time attitude control or geo-coding of image data, the on-board navigation system has to provide continuous dense output at equidistant time steps, which usually conflicts with the natural stepsize of the relevant integration methods and the non-equidistant measurement times. To cope with this problem, the integrator has to be supplemented by an interpolation scheme of compatible order and accuracy. After presenting a representative formulation of an on-board navigation system and deriving related timing and accuracy requirements, suitable Runge–Kutta methods and associated interpolants are selected and evaluated. Promising results are obtained for the classical RK4 method in combination with Richardson extrapolation and 5th-order Hermite interpolation. The 5th-order Fehlberg method with interpolation due to Enright and, for drag-free scenarios, the 5th-order Runge–Kutta–Nystrom method with 5th-order Hermite interpolation provide a good performance in terms of position interpolation. However, as both methods exhibit significant errors for the velocity interpolation, they are not recommended for use with the outlined navigation filter.

Stability-constrained model predictive control

This paper presents a new approach to guaranteeing stability of model predictive control (MPC). A stability constraint is computed and propagated forward at each stage as a constraint on the magnitude of the predicted state vector for a state-space controllable-form realization. It is shown that asymptotic stability is guaranteed for the case of constrained multi-input linear time-invariant systems under full-state measurement or with state estimation for any prediction horizon and any optimization objective function.