Two-Stage Extended Kalman Filters with Derivative-Free Local Linearizations

This paper proposes a derivative-free two-stage extended Kalman filter (2-EKF) especially suited for state and parameter identification of mechanical oscillators under Gaussian white noise. Two sources of modeling uncertainties are considered: (1)?errors in linearization, and (2) an inadequate system model. The state vector is presently composed of the original dynamical/parameter states plus the so-called bias states accounting for the unmodeled dynamics. An extended Kalman estimation concept is applied within a framework predicated on explicit and derivative-free local linearizations (DLL) of nonlinear drift terms in the governing stochastic differential equations (SDEs). The original and bias states are estimated by two separate filters; the bias filter improves the estimates of the original states. Measurements are artificially generated by corrupting the numerical solutions of the SDEs with noise through an implicit form of a higher-order linearization. Numerical illustrations are provided for a few single- and multidegree-of-freedom nonlinear oscillators, demonstrating the remarkable promise that 2-EKF holds over its more conventional EKF-based counterparts.     

On-Line Parametric Identification of MDOF Nonlinear Hysteretic Systems

A method based on adaptive estimation approaches is presented for the on-line identification of hysteretic systems under arbitrary dynamic environments. The availability of such an identification approach is crucial for the on-line control and monitoring of time-varying structural systems. Previous work by the writers is extended to handle the general case when no information is available on the system parameters, even the mass distribution. A robust, least-squares-based adaptive identification algorithm, incorporating a Bouc-Wen hysteresis element model with additional polynomial-type nonlinear terms, is used to investigate the effects of persistence of excitation and of under- and overparameterization: challenging problems in realistic applications. In spite of the challenges encountered in the identification of the hereditary nature of the restoring force of such nonlinear systems, it is shown through the use of simulation studies of single-degree-of-freedom and certain multi-degree-of-freedom systems that the proposed approach can yield reliable estimates of the hysteretic restoring force and the hysteretic element model parameters.     

Time Domain Identification of Frames under Earthquake Loadings

The purpose of this paper is to examine the adaptive identification method and the nonlinear system identification technique through numerical simulation on seismic response of building structures. For the identification of a time-variant system and a system with an abrupt change of modal parameters, two parameter adaptation algorithms in recursive identification methods are discussed: recursive least square algorithm with constant trace and adaptive fading Kalman filter method. Based on the equivalent linear model, the time-variant model parameters can be identified. The tracking ability on the identification of a time-variant system is discussed. For the identification of the nonlinear system, the forward including algorithm and the neural network training algorithm were used to estimate the modal parameter of the nonlinear autoregressive moving average model, and the accuracy on the predicted output of nonlinear model identification was discussed.     

Identification of Parametric Variations of Structures Based on Least Squares Estimation and Adaptive Tracking Technique

An important objective of health monitoring systems for civil infrastructures is to identify the state of the structure and to detect the damage when it occurs. System identification and damage detection, based on measured vibration data, have received considerable attention recently. Frequently, the damage of a structure may be reflected by a change of some parameters in structural elements, such as a degradation of the stiffness. Hence it is important to develop data analysis techniques that are capable of detecting the parametric changes of structural elements during a severe event, such as the earthquake. In this paper, we propose a new adaptive tracking technique, based on the least-squares estimation approach, to identify the time-varying structural parameters. In particular, the new technique proposed is capable of tracking the abrupt changes of system parameters from which the event and the severity of the structural damage may be detected. The proposed technique is applied to linear structures, including the Phase I ASCE structural health monitoring benchmark building, and a nonlinear elastic structure to demonstrate its performance and advantages. Simulation results demonstrate that the proposed technique is capable of tracking the parametric change of structures due to damages.     

A constrained neural network Kalman filter for price estimation in high frequency financial data

In this paper we present a neural network extended Kalman filter for modeling noisy financial time series. The neural network is employed to estimate the nonlinear dynamics of the extended Kalman filter. Conditions for the neural network weight matrix are provided to guarantee the stability of the filter. The extended Kalman filter presented is designed to filter three types of noise commonly observed in financial data: process noise, measurement noise, and arrival noise. The erratic arrival of data (arrival noise) results in the neural network predictions being iterated into the future. Constraining the neural network to have a fixed point at the origin produces better iterated predictions and more stable results. The performance of constrained and unconstrained neural networks within the extended Kalman filter is demonstrated on "Quote" tick data from the $/DM exchange rate (1993-1995).     

Investigation of a System Identification Methodology in the Context of the ASCE Benchmark Problem

This article briefly presents the theory for a system identification and damage detection algorithm for linear systems, and discusses the effectiveness of such a methodology in the context of a benchmark problem that was proposed by the ASCE Task Group in Health Monitoring. The proposed approach has two well-defined phases: (1) identification of a state space model using the Observer/Kalman filter identification algorithm, the eigensystem realization algorithm, and a nonlinear optimization approach based on sequential quadratic programming techniques, and (2) identification of the second-order dynamic model parameters from the realized state space model. Structural changes (damage) are characterized by investigating the changes in the second-order parameters of the “reference” and “damaged” models. An extensive numerical analysis, along with the underlying theory, is presented in order to assess the advantages and disadvantages of the proposed identification methodology.     

Least-Squares Estimation with Unknown Excitations for Damage Identification of Structures

System identification and damage detection for structural health monitoring of civil infrastructures have received considerable attention recently. Time domain analysis methodologies based on measured vibration data, such as the least-squares estimation and the extended Kalman filter, have been studied and shown to be useful. The traditional least-squares estimation method requires that all the external excitation data (input data) be available, which may not be the case for many structures. In this paper, a recursive least-squares estimation with unknown inputs (RLSE-UI) approach is proposed to identify the structural parameters, such as the stiffness, damping, and other nonlinear parameters, as well as the unmeasured excitations. Analytical recursive solutions for the proposed RLSE-UI are derived and presented. This analytical recursive solution for RLSE-UI is not available in the previous literature. An adaptive tracking technique recently developed is also implemented in the proposed approach to track the variations of structural parameters due to damages. Simulation results demonstrate that the proposed approach is capable of identifying the structural parameters, their variations due to damages, and unknown excitations.     

A model-based algorithm for blood glucose control in type I diabetic patients

A model-based predictive control algorithm is developed to maintain normoglycemia in the Type I diabetic patient using a closed-loop insulin infusion pump. Utilizing compartmental modeling techniques, a fundamental model of the diabetic patient is constructed. The resulting nineteenth-order nonlinear pharmacokinetic-pharmacodynamic representation is used in controller synthesis. Linear identification of an input-output model from noisy patient data is performed by filtering the impulse-response coefficients via projection onto the Laguerre basis. A linear model predictive controller is developed using the identified step response model. Controller performance for unmeasured disturbance rejection (50 g oral glucose tolerance test) is examined. Glucose setpoint tracking performance is improved by designing a second controller which substitutes a more detailed internal model including state-estimation and a Kalman filter for the input-output representation. The state-estimating controller maintains glucose within 15 mg/dl of the setpoint in the presence of measurement noise. Under noise-free conditions, the model-based predictive controller using state estimation outperforms an internal model controller from literature (49.4% reduction in undershoot and 45.7% reduction in settling time). These results demonstrate the potential use of predictive algorithms for blood glucose control in an insulin infusion pump.     

Improved Estimation of ADCP Apparent Bed-Load Velocity Using a Real-Time Kalman Filter

An estimate of apparent bed-load velocity (v) can be derived from the difference between differential global positioning system (DGPSs) and acoustic Doppler current profiler (ADCP) bottom track (BT) measurements when BT is biased by a moving bottom. A Kalman filter has been developed to integrate GPS and bottom track data to improve estimation of boat velocity during ADCP measurements under moving bed conditions (Rennie and Rainville, 2008, J. Hydraulic Eng., in review). The boat velocity estimated using the Kalman filter is superior to boat velocity from raw GPS data. In this paper we assess the improvement in estimation of v using the Kalman filter as opposed to raw GPS data. Specifically, a synthetic moving bed bias was generated for 22 repeat transects of the Gatineau River, Quebec. The synthetic moving bed bias had mean, variance, and distribution across the section as typically observed during bed-load transport conditions, and had the advantage that it was known explicitly. The errors in estimated apparent bed-load velocity derived using either raw DGPS data or the Kalman filter boat velocity were compared. It was found that the improved boat velocity from the Kalman filter yielded significantly (α = 0.05) better estimates of v, (e.g., 61% reduction in error when the Kalman filter boat velocity was used instead of wide area augmentation system GGA), because boat velocity errors were reduced. Tests with real moving bed data confirmed the Kalman filter was able to significantly reduce errors in bed load calculated with stand alone GPS.     

基于ELM和MCSCKF的锂离子电池SOC估计

为了减少噪声对锂离子电池荷电状态估计的影响,本文提出一种新颖的基于极限学习机和最大相关熵平方根容积卡尔曼滤波的SOC估计方法。首先,利用泛化性好、运行速度快的极限学习机作为卡尔曼滤波的测量方程;其次,基于灰狼优化算法,极限学习机的超参数被优化以提高电池荷电状态的估计精度;最后,基于最大相关熵平方根容积卡尔曼滤波,极限学习机的测量噪声被进一步减弱。所提方法可以简化极限学习机繁琐的调参过程,且为闭环的SOC估计方法。所提方法在多工况和宽温度范围内被测试以验证其泛化性能。测试结果显示,所提方法明显地提高了锂离子电池的荷电状态估计精度。同时,对比其他算法,所提方法的平均运行时间仅仅为长短时序列和循环门控单元网络的三分之一。当行驶工况复杂、温度变化区间较大时,所提方法的均方根误差小于1%,最大误差小于3%。当存在初始误差与环境噪声时,所提方法显示出了优越的鲁棒性。      

Improving Precision in the Reference Velocity of ADCP Measurements Using a Kalman Filter with GPS and Bottom Track

Global positioning system (GPS) data are used to measure boat velocity during acoustic Doppler current profiler (ADCP) discharge measurements, particularly when bottom tracking (BT) is biased by moving bed. A Kalman filter is developed to improve the velocity reference used by the ADCP under such conditions. Kalman filtering is a recursive statistical technique that estimates the current state of a process, given various inputs and their variance. In the case of data obtained by ADCP, the availability of two independent velocity measurements and a position measurement makes this method particularly attractive. The new Kalman filter combines raw inputs for GPS position (GGA) and Doppler velocity (VTG) with BT data in real time to produce best estimates of velocity. The technique is evaluated and calibrated using various accuracies of GPS data collected simultaneously along with unbiased BT data at two different sites. On the Gatineau River, real-time kinematic and wide area augmentation system corrections were used for this study. On the Saint Mary’s River, nondifferential GPS was collected. To examine the conditions under which such a system would be required, synthetic data for a moving bed contamination of BT were created. In all moving bed conditions evaluated, the Kalman filter estimates of reference velocity were superior to raw inputs.     

Real-Time Demand Estimation and Confidence Limit Analysis for Water Distribution Systems

A real-time estimation of water distribution system state variables such as nodal pressures and chlorine concentrations can lead to savings in time and money and provide better customer service. While a good knowledge of nodal demands is prerequisite for pressure and water quality prediction, little effort has been placed in real-time demand estimation. This study presents a real-time demand estimation method using field measurement provided by supervisory control and data acquisition systems. For real-time demand estimation, a recursive state estimator based on weighted least-squares scheme and Kalman filter are applied. Furthermore, based on estimated demands, real-time nodal pressures and chlorine concentrations are predicted. The uncertainties in demand estimates and predicted state variables are quantified in terms of confidence limits. The approximate methods such as first-order second-moment analysis and Latin hypercube sampling are used for uncertainty quantification and verified by Monte Carlo simulation. Application to a real network with synthetically generated data gives good demand estimations and reliable predictions of nodal pressure and chlorine concentration. Alternative measurement data sets are compared to assess the value of measurement types for demand estimation. With the defined measurement error magnitudes, pipe flow data are significantly more important than pressure head measurements in estimating demands with a high degree of confidence.     

Subsurface Radioactive Contaminant Transport Modeling Using Particle and Kalman Filter Schemes

Contamination of groundwater by radioactive contaminants can be harmful to the environment. Various prediction models have been adopted to simulate the state of contaminants in the subsurface. Conventional numerical models are simplified by approximation and the model parameters are assumed to be constant, thereby introducing error to the prediction results. Particle and Kalman filters are used in this research to simulate the radioactive contaminant cobalt-57 transport in a subsurface environment by using a two-dimensional advection-dispersion model. A radioactive contaminant concentration was predicted spatially and temporally within boundary conditions. The errors in the prediction results were assessed by using the root-mean-square-error (RMSE) equation. The results show that the Kalman filter performs better than the particle filter when the prediction model is linear. Furthermore, the results from filters are closer to the true value in comparison with the numerical solution, and the filters are capable of reducing the RMSE of the numerical solution by approximately 80%.     

基于快速SR-UKF的锂离子动力电池SOC联合估计

针对标准无迹卡尔曼滤波(Unscented Kalman filter, UKF) 算法本身存在着因状态误差协方差矩阵无法实现Cholesky分解而导致滤波发散的隐患,以及在电池状态估计过程中由离线标定的电池等效模型参数而造成的累积误差的问题,本文发展了一种平方根无迹卡尔曼滤波(Square-root unscented Kalman filter, SR-UKF)算法,并设计了一种电池状态联合估计策略。首先快速SR-UKF算法通过对观测方程进行准线性化处理,降低了每次无迹变换时的计算开销;然后在迭代过程中,用状态误差协方差矩阵的平方根代替状态误差协方差矩阵,该平方根是由QR分解与 Cholesky因子的一阶更新得到,解决了UKF 算法迭代过程中可能由计算累积误差引起状态误差协方差矩阵负定而导致滤波结果发散的问题,保证了电池荷电状态(State of charge,SOC)在线滚动估计的数值稳定性;最后采用联合估计策略,对电池等效模型参数进行实时辨识,保证了电池等效模型的准确性与有效性,从而提高了电池SOC的估计精度。仿真对比结果验证了快速SR-UKF算法以及电池状态联合估计策略的可行性与鲁棒性。      

Structural Reliability Estimation with Vibration-Based Identified Parameters

This paper presents a unique structural reliability estimation method incorporating structural parameter identification results based on the seismic response measurement. In the shaking table test, a three-bent concrete bridge model was shaken to different damage levels by a sequence of earthquake motions with increasing intensities. Structural parameters, stiffness and damping values of the bridge were identified under damaging seismic events based on the seismic response measurement. A methodology was developed to understand the importance of structural parameter identification in the reliability estimation. Along this line, a set of structural parameters were generated based on the Monte Carlo simulation. Each of them was assigned to the base bridge model. Then, every bridge model was analyzed using nonlinear time history analyses to obtain damage level at the specific locations. Last, reliability estimation was performed for bridges modeled with two sets of structural parameters. The first one was obtained by the nonlinear time history analysis with the Monte Carlo simulated parameters which is called nonupdated structural parameters. The second one was obtained by updating the first set in Bayesian sense based on the vibration-based identification results which is called updated structural parameters. In the scope of this paper, it was shown that residual reliability of the system estimated using the updated structural parameters is lower than the one estimated using the nonupdated structural parameters.     

Cooperative Navigation for Autonomous Underwater Vehicles Based on Estimation of Motion Radius Vectors

A cooperative navigation algorithm for a group of autonomous underwater vehicles is proposed on the basis of mo-tion radius vector estimation. Combined the dead reckoning data with the mutual range data through an acoustic communica-tion network among the group members, the relative positioning problem can be solved. A novel approach for solving the relative positioning is presented by using a recursive trigonometry technique and extended Kalman filter(EKF). Simulation results verify the correctness and effectiveness of this navigation method.     

Extended Kalman Filter-Based Pruning Method for Recurrent Neural Networks

Pruning is one of the effective techniques for improving the generalization error of neural networks. Existing pruning techniques are derived mainly from the viewpoint of energy minimization, which is commonly used in gradient-based learning methods. In recurrent networks, extended Kalman filter (EKF)-based training has been shown to be superior to gradient-based learning methods in terms of speed. This article explains a pruning procedure for recurrent neural networks using EKF training. The sensitivity of a posterior probability is used as a measure of the importance of a weight instead of error sensitivity since posterior probability density is readily obtained from this training method. The pruning procedure is tested using three problems: (1) the prediction of a simple linear time series, (2) the identification of a nonlinear system, and (3) the prediction of an exchange-rate time series. Simulation results demonstrate that the proposed pruning method is able to reduce the number of parameters and improve the generalization ability of a recurrent network.     

Covariate measurement error and the estimation of random effect parameters in a mixed model for longitudinal data

We explore the effects of measurement error in a time-varying covariate for a mixed model applied to a longitudinal study of plasma levels and dietary intake of beta-carotene. We derive a simple expression for the bias of large sample estimates of the variance of random effects in a longitudinal model for plasma levels when dietary intake is treated as a time-varying covariate subject to measurement error. In general, estimates for these variances made without consideration of measurement error are biased positively, unlike estimates for the slope coefficients which tend to be 'attenuated'. If we can assume that the residuals from a longitudinal fit for the time-varying covariate behave like measurement errors, we can estimate the original parameters without the need for additional validation or reliability studies. We propose a method to test this assumption and show that the assumption is reasonable for the example data. We then use a likelihood-based method of estimation that involves a simple extension of existing methods for fitting mixed models. Simulations illustrate the properties estimators.     

Structural Health Monitoring by Recursive Bayesian Filtering

A new vision of structural health monitoring (SHM) is presented, in which the ultimate goal of SHM is not limited to damage identification, but to describe the structure by a probabilistic model, whose parameters and uncertainty are periodically updated using measured data in a recursive Bayesian filtering (RBF) approach. Such a model of a structure is essential in evaluating its current condition and predicting its future performance in a probabilistic context. RBF is conventionally implemented by the extended Kalman filter, which suffers from its intrinsic drawbacks. Recent progress on high-fidelity propagation of a probability distribution through nonlinear functions has revived RBF as a promising tool for SHM. The central difference filter, as an example of the new versions of RBF, is implemented in this study, with the adaptation of a convergence and consistency improvement technique. Two numerical examples are presented to demonstrate the superior capacity of RBF for a SHM purpose. The proposed method is also validated by large-scale shake table tests on a reinforced concrete two-span three-bent bridge specimen.     

Aircraft Icing Detection, Identification, and Reconfigurable Control Based on Kalman Filtering and Neural Networks

Flight in all weather conditions has necessitated correctly detecting icing and taking reasonable measures against it. This work aims at the detection and identification of airframe icing based on statistical properties of aircraft dynamics and reconfigurable control protecting aircraft from hazardous icing conditions. A Kalman filter is used for the data collection for the detection of icing, which aerodynamically deteriorates flight performance. A neural network process is applied for the identification of icing model of the aircraft, which is represented by five parameters based on past experiments for iced wing airfoils. Icing is detected by a Kalman filtering innovation sequence approach. A neural network structure is embodied such that its inputs are the aircraft estimated measurements and its outputs are the parameters affected by ice, which corresponds to the aircraft inverse dynamic model. The necessary training and validation set for the neural network model of the iced aircraft are obtained from the simulations of nominal model, which are performed for various icing conditions. In order to decrease noise effects on the states and to increase training performance of the neural network, the estimated states by the Kalman filter are used. A suitable neural network model of aircraft inverse dynamics is obtained by using system identification methods and learning algorithms. This trained model is used as an application for the control of the aircraft that has lost its controllability due to icing. The method is applied to F16 military and A340 commercial aircraft models and the results seem to be good enough.