Optimal Two-stage Kalman Filter Research Articles

Inverse estimation of heat flux and temperature field for a nonlinear heat transfer system using step-renewed two-stage Kalman filter

Inverse estimation of the thermal input and states is a useful tool for the monitoring of a thermal system. Nonlinear heat transfer is a quite common issue in this kind of problem. As the thermal properties are temperature-dependent, it is difficult to inversely estimate the input and temperature field. To deal with this problem, a step-renewed scheme is adopted into the weighted optimal two-stage Kalman filter (WOTSKF), forming step-renewed WOTSKF (SR-WOTSKF). This strategy is a marching method, without solving the Jacobian matrix in extended Kalman filter. The feasibility of dealing with the nonlinear inverse heat transfer problem is implemented by comparing WOTSKF and SR-WOTSKF. Then, the weighting factor, initial guess of the input heat flux and measurement noise are changed to investigate their effects on the simultaneous estimation results of heat flux and temperature field. It is found that when the weighting factor is 0.95, the relative error of the estimated heat flux using SR-WOTSKF is approximately 2.73% which is much lower than that using WOTSKF. Results using SR-WOTSKF demonstrate that the estimation results are independent of the initial guess and the estimated heat flux can track the exact one in a short period. An incorrect initial guess causes the initial estimation error of the temperature and heat flux and it makes no sense to following period. When the standard deviations of the measurement noise are adopted as 0.01, 0.10, 0.50 and 1.00, the mean relative error of the heat flux is 2.73%, 4.79%, 7.71% and 9.43%, demonstrating the strong robustness of SR-WOTSKF to resist the measurement noise.

Simultaneous estimation of heat flux and temperature field using optimal two-stage Kalman filters

Inverse estimation of unknown input heat flux and temperature field plays an important role in thermal state monitoring. Optimal two-stage Kalman filter (OTSKF) is introduced into the inverse estimation of thermal state in this work. To further improve its performance, a new method weighted optimal two-stage Kalman filter (WOTSKF) algorithm is developed, in which a forgetting factor is used to modify OTSKF. The estimation performances of OTSKF and WOTSKF are compared in a two-dimensional heat transfer system. They are employed to estimate the state with heat flux of temporal, spatial, and spatiotemporal variation. Additionally, the effects of forgetting factor and measurement noises on the estimation accuracy of heat flux and temperature are investigated. Though decreasing the forgetting factor can improve the tracking ability and estimation accuracy of WOTSKF, it should not be too low due to the reason that low α causes fluctuation of estimation results. When the standard deviation of measurement noises σr increases from 0.05 to 2.00, the mean relative error of estimated heat flux ηq using OTSKF rises from 4.03% to 10.67% and it climbs from 2.76% to 7.07% using WOTSKF. That demonstrates proposed WOTSKF has stronger robustness than OTSKF.

A Two-stage Kalman Filter for Cyber-attack Detection in Automatic Generation Control System

Communication plays a vital role in incorporating smartness into the interconnected power system. However, historical records prove that the data transfer has always been vulnerable to cyber-attacks. Unless these cyber-attacks are identified and cordoned off, they may lead to black-out and result in national security issues. This paper proposes an optimal two-stage Kalman filter (OTS-KF) for simultaneous state and cyber-attack estimation in automatic generation control (AGC) system. Biases/cyber-attacks are modeled as unknown inputs in the AGC dynamics. Five types of cyber-attacks, i.e., false data injection (FDI), data replay attack, denial of service (DoS), scaling, and ramp attacks, are injected into the measurements and estimated using OTS-KF. As the load variations of each area are seldom available, OTS-KF is reformulated to estimate the states and outliers along with the load variations of the system. The proposed technique is validated on the benchmark two-area, three-area, and five-area power system models. The simulation results under various test conditions demonstrate the efficacy of the proposed filter.

Aircraft Inertial Measurement Unit Fault Diagnosis Based on Optimal Two-Stage UKF

Optimal two stage Kalman filter (OTSKF) is able to obtain optimal estimation of system states and bias for linear system which contains random bias. Unscented Kalman filter (UKF) is a conventional nonlinear filtering method which utilizes Sigmas point sampling and unscented transformation technology realizes propagation of state means and covariances through nonlinear system. Aircraft is a typical complicate nonlinear system, this paper treats the faults of Inertial Measurement Unit (IMU) as random bias, established a filtering model which contains faults of IMU. Hybird the two stage filtering technique and UKF, this paper proposed an optimal two stage unscented Kalman filter (OTSUKF) algorithm which is suitable for fault diagnosis of IMU, realized optimal estimation of system states and faults identification of IMU via proposed innovative designing method of filtering model and the algorithm was validated that it is robust to wind disterbance via real flight data and it is also validated that proposed OTSUKF is optimal in the existance of wind disturbance via comparing with the existance iterated optimal two stage extended kalman filter (IOTSEKF) method.

Direct Torque Control of Sensorless Induction Machine Drives: A Two-Stage Kalman Filter Approach

Extended Kalman filter (EKF) has been widely applied for sensorless direct torque control (DTC) in induction machines (IMs). One key problem associated with EKF is that the estimator suffers from computational burden and numerical problems resulting from high order mathematical models. To reduce the computational cost, a two-stage extended Kalman filter (TEKF) based solution is presented for closed-loop stator flux, speed, and torque estimation of IM to achieve sensorless DTC-SVM operations in this paper. The novel observer can be similarly derived as the optimal two-stage Kalman filter (TKF) which has been proposed by several researchers. Compared to a straightforward implementation of a conventional EKF, the TEKF estimator can reduce the number of arithmetic operations. Simulation and experimental results verify the performance of the proposed TEKF estimator for DTC of IMs.

High Maneuvering Multiple-Underwater Robot Tracking with Optimal Two-Stage Kalman Filter and Competitive Hopfield Neural Network Based Data Fusion

Tracking high maneuvering underwater targets with radar is very complex, because the acceleration is not directly measurable. This may cause the tracking errors which will diverge the estimation when maneuver happened. In this article Optimal Two-Stage Kalman Filter (OTSKF) used for tracking purpose, which can estimate the acceleration in maneuvering targets. Under this method, Acceleration of the targets has been identified and it is estimated at the moment. Also an effective data fusion method plays critical role in order to track the objects accurately. We use Competitive Hopfield Neural Network (CHNN) to associate obtained data from radar measurements with existing target. In this paper we investigate new method, which combine these two algorithms for tracking highly maneuvering targets. As simulation results show, to solve multiple-underwater vehicles tracking problem, investigated method have good ability and suitable performance in high maneuvering targets.

A novel robust two-stage extended Kalman filter for bearings-only maneuvering target tracking

A novel robust filter is proposed via embedding extended Kalman filter into an optimal two-stage Kalman filter, which is improved through a modified Sage-Husa noise statistics estimator. Meanwhile, strong tracking multiple fading factors are introduced in this paper to improve the tracking performance for high maneuvering targets. The new method provides an optimal estimation of the target state through a combination of the output of the first stage (a “bias-free” filter) and that of the second stage (a “bias-compensating” filter). Furthermore, the unknown statistical parameters of virtual noises are estimated online, and the predicted covariance can be adjusted in real time by fading factors when high maneuvers occur. Simulation results of tracking a maneuvering target by 3 passive sensors show that the proposed algorithm has advantages over the conventional method in terms of the numerical accuracy with only a little additional computational cost. Key words: Target tracking, two-stage Kalman filter, noise statistics estimator, multiple fading factors.

Adaptive Two-Stage Extended Kalman Filter for a Fault-Tolerant INS-GPS Loosely Coupled System

The well-known conventional Kalman filter requires an accurate system model and exact stochastic information. But in a number of situations, the system model has an unknown bias, which may degrade the performance of the Kalman filter or may cause the filter to diverge. The effect of the unknown bias may be more pronounced on the extended Kalman filter (EKF), which is a nonlinear filter. The two-stage extended Kalman filter (TEKF) with respect to this problem has been receiving considerable attention for a long time. Recently, the optimal two-stage Kalman filter (TKF) for linear stochastic systems with a constant bias or a random bias has been proposed by several researchers. A TEKF can also be similarly derived as the optimal TKF. In the case of a random bias, the TEKF assumes that the information of a random bias is known. But the information of a random bias is unknown or partially known in general. To solve this problem, this paper proposes an adaptive two-stage extended Kalman filter (ATEKF) using an adaptive fading EKF. To verify the performance of the proposed ATEKF, the ATEKF is applied to the INS-GPS (inertial navigation system-Global Positioning System) loosely coupled system with an unknown fault bias. The proposed ATEKF tracked/estimated the unknown bias effectively although the information about the random bias was unknown.

A Unified Solution to Unbiased Minimum-Variance Estimation for Systems with Unknown Inputs

A parameterized three-stage Kalman filter (PTSKF) is proposed, serving as a unified solution to unbiased minimum-variance estimation for systems with unknown inputs that affect both the system and the outputs. The PTSKF is characterized by two design parameters and includes three parts: one is for the main system state estimate, the second is for the optimal unknown inputs estimate, and the last is added to further enhance the robust filtering performance. It is shown that the extended robust two-stage Kalman filter (ERTSKF), which is an extension of the previously proposed RTSKF, and the optimal two-stage Kalman filter (OTSKF) are special cases of this new filter. Simulation results show that not only the filtering performance of the PTSKF is compatible to that of the previous proposed parameterized minimum-variance filter (PMVF) but also the computational complexity of the former is less intensive than that of the latter.

Two-stage Kalman filtering via structured square-root

This paper considers the problem of estimating an unknown input (bias) by means of the augmented-state Kalman (AKF) filter. To reduce the computational complexity of the AKF, [12] recently developed an optimal two-stage Kalman filter (TS-AKF) that separates the bias estimation from the state estimation, and shows that his new two-stage estimator is equivalent to the standard AKF, but requires less computations per iteration. This paper focuses on the derivation of the optimal two-stage estimator for the square-root covariance implementation of the Kalman filter (TS-SRCKF), which is known to be numerically more robust than the standard covariance implementation. The new TS-SRCKF also estimates the state and the bias separately while at the same time it remains equivalent to the standard augmented-state SRCKF. It is experimentally shown in the paper that the new TS-SRCKF may require less flops per iteration for some problems than the Hsieh's TS-AKF [12]. Furthermore a second, even faster (single-stage) algorithm has been derived in the paper by exploiting the structure of the least-squares problem and the square-root covariance formulation of the AKF. The computational complexities of the two proposed methods have been analyzed and compared the those of other existing implementations of the AKF.

Separate-bias estimation with reduced-order Kalman filters

This paper presents the optimal two-stage Kalman filter for systems that involve noise-free observations and constant but unknown bias. Like the full-order separate-bias Kalman filter, this new filter provides an alternative to state vector augmentation and offers the same potential for improved numerical accuracy and reduced computational burden. When dealing with systems involving accurate, essentially noise-free measurements, this new filter offers an additional advantage, a reduction in filter order. The optimal separate-bias reduced order estimator involves a reduced order filter for estimating the state, the order equalling the number of states less the number of observations.

Optimal two-stage Kalman filter in the presence of random bias

This paper gives an optimal solution of the two-stage Kalman filter for linear stochastic systems subject to random bias. It is shown that the state estimate can be expressed as x k k = x ̃ k k + β k k b k k where x ̃ k k is a modified bias-free state estimate and b k k the optimal estimate of random bias.