Unscented Transformation Based Robust Kalman Filter and Its Applications in Fermentation Process

State estimation is the precondition and foundation of a bioprocess monitoring and optimal control. However,there are many difficulties in dealing with a non-linear system,such as the instability of process, un-modeled dynamics,parameter sensitivity,etc.This paper discusses the principles and characteristics of three different approaches,extended Kalman filters,strong tracking filters and unscented transformation based Kalman filters.By introducing the unscented transformation method and a sub-optimal fading factor to correct the prediction error covariance,an improved Kalman filter,unscented transformation based robust Kalman filter,is proposed. The performance of the algorithm is compared with the strong tracking filter and unscented transformation based Kalman filter and illustrated in a typical case study for glutathione fermentation process.The results show that the proposed algorithm presents better accuracy and stability on the state estimation in numerical calculations.

Unscented extended Kalman filter for target tracking

A new method of unscented extended Kalman filter （UEKF） for nonlinear system is presented. This new method is a combination of the unscented transformation and the extended Kalman filter （EKF）. The extended Kalman filter is similar to that in a conventional EKF. However, in every running step of the EKF the unscented transformation is running, the deterministic sample is caught by unscented transformation, then posterior mean of non- lineadty is caught by propagating, but the posterior covariance of nonlinearity is caught by linearizing. The accuracy of new method is a little better than that of the unscented Kalman filter （UKF）, however, the computational time of the UEKF is much less than that of the UKF.

Major Development Under Gaussian Filtering Since Unscented Kalman Filter

Filtering is a recursive estimation of hidden states of a dynamic system from noisy measurements.Such problems appear in several branches of science and technology,ranging from target tracking to biomedical monitoring.A commonly practiced approach of filtering with nonlinear systems is Gaussian filtering.The early Gaussian filters used a derivative-based implementation,and suffered from several drawbacks,such as the smoothness requirements of system models and poor stability.A derivative-free numerical approximation-based Gaussian filter,named the unscented Kalman filter(UKF),was introduced in the nineties,which offered several advantages over the derivativebased Gaussian filters.Since the proposition of UKF,derivativefree Gaussian filtering has been a highly active research area.This paper reviews significant developments made under Gaussian filtering since the proposition of UKF.The review is particularly focused on three categories of developments:i)advancing the numerical approximation methods;ii)modifying the conventional Gaussian approach to further improve the filtering performance;and iii)constrained filtering to address the problem of discrete-time formulation of process dynamics.This review highlights the computational aspect of recent developments in all three categories.The performance of various filters are analyzed by simulating them with real-life target tracking problems.

Application of Unscented Kalman Filter in Satellite Orbit Simulation

A new estimate method is proposed, which takes advantage of the unscented transform method, thus the true mean and covariance are approximated more accurately. The new method can be applied to non-linear systems without the linearization process necessary for the EKF, and it does not demand a Gaussian distribution of noise and what’s more, its ease of implementation and more accurate estimation features enables it to demonstrate its good performance in the experiment of satellite orbit simulation. Numerical experiments show that the application of the unscented Kalman filter is more effective than the EKF.

Robust Iterated Sigma Point FastSLAM Algorithm for Mobile Robot Simultaneous Localization and Mapping

Simultaneous localization and mapping （SLAM） is a key technology for mobile robots operating under unknown environment. While FastSLAM algorithm is a popular solution to the SLAM problem, it suffers from two major drawbacks： one is particle set degeneracy due to lack of observation information in proposal distribution design of the particle filter; the other is errors accumulation caused by linearization of the nonlinear robot motion model and the nonlinear environment observation model. For the purpose of overcoming the above problems, a new iterated sigma point FastSLAM （ISP-FastSLAM） algorithm is proposed. The main contribution of the algorithm lies in the utilization of iterated sigma point Kalman filter （ISPKF）, which minimizes statistical linearization error through Gaussian-Newton iteration, to design an optimal proposal distribution of the particle filter and to estimate the environment landmarks. On the basis of Rao-Blackwellized particle filter, the proposed ISP-FastSLAM algorithm is comprised by two main parts： in the first part, an iterated sigma point particle filter （ISPPF） to localize the robot is proposed, in which the proposal distribution is accurately estimated by the ISPKF; in the second part, a set of ISPKFs is used to estimate the environment landmarks. The simulation test of the proposed ISP-FastSLAM algorithm compared with FastSLAM2.0 algorithm and Unscented FastSLAM algorithm is carried out, and the performances of the three algorithms are compared. The simulation and comparing results show that the proposed ISP-FastSLAM outperforms other two algorithms both in accuracy and in robustness. The proposed algorithm provides reference for the optimization research of FastSLAM algorithm.

Optimal and suboptimal white noise smoothers for nonlinear stochastic systems

A new approach of smoothing the white noise for nonlinear stochastic system was proposed. Through presenting the Gaussian approximation about the white noise posterior smoothing probability density fimction, an optimal and unifying white noise smoothing framework was firstly derived on the basis of the existing state smoother. The proposed framework was only formal in the sense that it rarely could be directly used in practice since the model nonlinearity resulted in the intractability and infeasibility of analytically computing the smoothing gain. For this reason, a suboptimal and practical white noise smoother, which is called the unscented white noise smoother （UWNS）, was further developed by applying unscented transformation to numerically approximate the smoothing gain. Simulation results show the superior performance of the proposed UWNS approach as compared to the existing extended white noise smoother （EWNS） based on the first-order linearization.

Dynamic State Estimation of Power Systems with Uncertainties Based on Robust Adaptive Unscented Kalman Filter

In this paper,a robust adaptive unscented Kalman filter(RAUKF)is developed to mitigate the unfavorable effects derived from uncertainties in noise and in the model.To address these issues,a robust M-estimator is first utilized to update the measurement noise covariance.Next,to deal with the effects of model parameter errors while considering the computational complexity and real-time requirements of dynamic state estimation,an adaptive update method is produced.The proposed method is integrated with spherical simplex unscented transformation technology,and then a novel derivative-free filter is proposed to dynamically track the states of the power system against uncertainties.Finally,the effectiveness and robustness of the proposed method are demonstrated through extensive simulation experiments on an IEEE 39-bus test system.Compared with other methods,the proposed method can capture the dynamic characteristics of a synchronous generator more reliably.

Consecutive tracking for ballistic missile based on bearings-only during boost phase

This paper proposes a modified centralized shifted Rayleigh filter（MCSRF） algorithm for tracking boost phase of ballistic missile（BM） trajectory with a highly nonlinear dynamical model based on bearings-only.This paper contributes three folds.Firstly,the mathematical model of an MCSRF for multiple passive sensors is derived.Then,minimum entropy based onedimensional optimization search to adaptively adjust the probability of the different filters for real time state estimation is deployed.Finally,the unscented transform（UT） is introduced to resolve the asymmetric state estimation problem.Simulation results show that the proposed algorithm can consecutively track the BM precisely during the boost phase.In comparison with the unscented Kalman filter（UKF） algorithm,the proposed algorithm effectively reduces the tracking position and velocity root mean square（RMS） errors,which will make more sense for early precision interception.

An image segmentation based algorithm for imaging of slow slip earthquakes

Laboratory experiments next to a variety of observations, especially in subduction zones, have explored the existence of a premonitory stable slow slip growth phase preceding large earthquakes. These phe- nomena play an important role in the earthquake cycle and thus precise imaging and monitoring of these events are of great significance. In the literature, ENIF （extended network inversion filter） has been proposed as a rigorous algorithm capable of isolating signal from different types of noise and thereby provides us with deep insight into spatio-temporal evolution of slow slip events. Despite its considerable advantages, ENIF still suffers from some limitations. ENIF applies Tikhonov method of regularization with a quadratic form of cost function. While anomalous slip regions have clear contrast with the background slip in reality, Tikhonov regularization tends to over smooth （globally smooth） the slipping portion on the estimated images. In order to avoid over smoothing phenomenon, we have incorporated into ENIF an image segmentation step which tries to preserve edges of slow-slip event. As a second limitation, due to the nonlinearity imposed by such constraint as non-negativity of slip rate, uncertainty propagation through model is not simple. As the core of ENIF, EKF （extended Kalman filter）, performs uncertainty propagation by linearization of nonlinear model using Jacobian and Hessian matrices. As an alternative for EKE we have also investigated the application of UKF （unscented Kalman filter） which uses UT （unscented transform） for uncertainty propagation. Finally, we tested our proposed algorithm using a low signal to noise ratio synthetic data set. The results show a significant improvement in the perfor- mance of ENIF when the segmentation step is incorporated into the algorithm.

Reliability analysis based on a novel density estimation method for structures with correlations

Estimating the Probability Density Function（PDF） of the performance function is a direct way for structural reliability analysis,and the failure probability can be easily obtained by integration in the failure domain.However,efficiently estimating the PDF is still an urgent problem to be solved.The existing fractional moment based maximum entropy has provided a very advanced method for the PDF estimation,whereas the main shortcoming is that it limits the application of the reliability analysis method only to structures with independent inputs.While in fact,structures with correlated inputs always exist in engineering,thus this paper improves the maximum entropy method,and applies the Unscented Transformation（UT） technique to compute the fractional moments of the performance function for structures with correlations,which is a very efficient moment estimation method for models with any inputs.The proposed method can precisely estimate the probability distributions of performance functions for structures with correlations.Besides,the number of function evaluations of the proposed method in reliability analysis,which is determined by UT,is really small.Several examples are employed to illustrate the accuracy and advantages of the proposed method.

Analysis of quantization noise and state estimation with quantized measurements

The approximate correction of the additive white noise model in quantized Kalman filter is investigated under certain conditions. The probability density function of the error of quantized measurements is analyzed theoretically and experimentally. The analysis is based on the probability theory and nonparametric density estimation technique, respectively. The approximator of probability density function of quantized measurement noise is given. The numerical results of nonparametric density estimation algorithm demonstrate that the theoretical conclusion is reasonable. Based on the analysis of quantization noise, a novel algorithm for state estimation with quantized measurements also is proposed. The algorithm is based on the least-squares estimator and unscented transform. By least-squares estimator, the effective information is extracted from the quantized measurements. Also, using the information to update the estimated state can give a better estimation under the influence of quantization. The root mean square error (RMSE) of the proposed algorithm is compared with the RMSE of the existing methods for a typical tracking scenario in wireless sensor networks systems. Simulations provide a strong evidence that this tracking algorithm could indeed give us a more precise estimated result.