A novel approach using parallel Kalman filters in time and frequency estimation

In the present information based knowledge society, accurate knowledge of time and frequency plays a fundamental role. For many applications, real time estimates of the clocks time and frequency error are required. We have developed a novel approach for estimating time and frequency errors based on an assembly of clocks of varying quality. By using parallel Kalman filters we utilize all available measurements in the estimation. As these measurements are available with different delays, the Kalman filter produces estimates of different quality. One filtermay produce real-time estimates while another filter waits for delayed measurements. When new information becomes available, the parallel Kalman filters exchange information in order to keep the state matrices updated with the most recent information. In Kalman filtering, accurate modelling of the measurement system is fundamental. All the contributing clocks, as well as the time transfer methods, are modelled as stochastic processes. By using this methodology and by correctly modelling the contributing clocks, we obtain minimum mean square error estimators (MMSE) of the time and frequency errors of a specific clock at every epoch. In addition, we also determine the uncertainty of each time and frequency error estimate. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of three types of piezoelectric cantilever on the topography quality in the vicinity of rough surface in a fluid ambient

The use of higher vibration modes and different geometries of the atomic force microscopy (AFM) piezoelectric micro cantilever (MC) is affected by the surface topography quality in a liquid medium. Therefore, utilizing an appropriate MC geometry and vibration mode is of a great importance. This paper analyzes the effect of different types of AFM MCs on the surface topography quality in the noncontact and tapping modes in a liquid medium. The modified couple stress theory (MCS) in a liquid based on the Timoshenko beam theory is used in order to enhance the accuracy of MC dynamic modeling. In addition, the differential quadrature (DQ) method has been used to discrete the equations. Identification of environmental forces helps to measure the accurate MC vibration amplitude. Investigating the effect of geometric and force parameters on the MC vibration behavior leads to understanding the system to design it properly in a liquid medium. Based on the advanced dynamic modeling, the best MC geometry for the specific surface roughness has been determined in the liquid for the surface topography. Also, due to oscillating the MC near the sample surface, the effect of interaction forces between the sample surface and the MC, including van der Waals, contact and squeeze forces is analyzed in a liquid medium in addition to the hydrodynamic forces. Furthermore, due to the reduction of the MC amplitude caused by the squeeze force; the MC is angled in proportion to the horizontal surface.     

A simple smoothing spline,III

Summary  Computational methods for spline smoothing are studied in the context of the linear smoothing spline. Comparisons are made between two efficient methods for computing the estimator using band-limited basis functions and the Kalman filter. In particular, the Kalman filter approach is shown to be an efficient method for computing under the Kimeldorf-Wahba representation for the estimator. Run time comparisons are made between band-limited B-spline and Kalman filter based algorithms.     

A low-order active fault-tolerant state space self-tuner for the unknown sampled-data nonlinear singular system using OKID and modified ARMAX model-based system identification

In this paper, we present two control schemes for the unknown sampled-data nonlinear singular system. One is an observer-based digital redesign tracker with the state-feedback gain and the feed-forward gain based on off-line observer/Kalman filter identification (OKID) method. The presented control scheme is able to make the unknown sampled-data nonlinear singular system to well track the desired reference signal. The other is an active fault tolerance state-space self-tuner using the OKID method and modified autoregressive moving average with exogenous inputs (ARMAX) model-based system identification for unknown sampled-data nonlinear singular system with input faults. First, one can apply the off-line OKID method to determine the appropriate (low-) order of the unknown system order and good initial parameters of the modified ARMAX model to improve the convergent speed of recursive extended-least-squares (RELS) method. Then, based on modified ARMAX-based system identification, a corresponding adaptive digital control scheme is presented for the unknown sampled-data nonlinear singular system with immeasurable system state. Moreover, in order to overcome the interference of input fault, one can use a fault-tolerant control scheme for unknown sampled-data nonlinear singular system by modifying the conventional self-tuner control (STC). The presented method can effectively cope with partially abrupt and/or gradual system input faults. Finally, some illustrative examples including a real circuit system are given to demonstrate the effectiveness of the presented design methodologies.     

基于方根分解形式的带衰减因子UKF算法应用

UKF作为一种新的非线性滤波方法已在目标跟踪问题中得到应用,在状态的时间更新阶段直接使用非线性模型,不引入线性化误差,而且不必计算Jacobians矩阵.提出了一种基于方根分解形式的带有衰减因子的UKF算法(SRDMA-UKF),算法的方根形式增加了数字稳定性和状态协方差的半正定性.通过衰减因子的引入加强对当前测量数据的利用,减小历史数据对滤波的影响.仿真实验结果表明,该算法与UKF算法相比具有更好的滤波性能.     

On the fractional-order extended Kalman filter and its application to chaotic cryptography in noisy environment

In this paper via a novel method of discretized continuous-time Kalman filter, the problem of synchronization and cryptography in fractional-order systems has been investigated in presence of noisy environment for process and output signals. The fractional-order Kalman filter equation, applicable for linear systems, and its extension called the extended Kalman filter, which can be used for nonlinear systems, are derived. The result is utilized for chaos synchronization with the aim of cryptography while the transmitter system is fractional-order, and both the transmitter and transmission channel are noisy. The fractional-order stochastic chaotic Chen system is then presented to apply the proposed method for chaotic signal cryptography. The results show the effectiveness of the proposed method.     

A computationally efficient Kalman smoother for the evaluation of the CH4 budget in Europe

Two data assimilation methods of the Kalman filtering approach are applied to the evaluation of the methane (CH₄) distribution in the atmosphere of Europe. The long term historical observation data of CH₄ concentration are integrated with the dynamical Eulerian dispersion model (EUROS). In each proposed method a specific algorithm is employed to avoid the heavy computation burden and huge storage requirement of the conventional Kalman filter for large scale systems. Moreover, a smoother algorithm is developed to identify the emission input. The feasibility of proposed data assimilation methods is verified by the application results.     

Forecasting Using Growth Curves—An Adaptive Approach

Growth curves such as the logistic and Gompertz are widely used for forecasting market development. The approach proposed is specifically designed for forecasting, rather than fitting available data—the usual approach with non-linear least squares regression. Two innovations form the foundation for this approach. The growth curves are reformulated from a time basis to an observation basis. This ensures that the available observations and the forecasts form a monotonic series; this is not necessarily true for least squares extrapolations of growth curves. An extension of the Kalman filter, an approach already used with linear forecasting models, is applied to the estimation of the growth curve coefficients. This allows the coefficients the flexibility to change over time if the market environment changes. The extended Kalman filter also proves the information for the generation of confidence intervals about the forecasts. Alternative forecasting approaches, least squares and an adaptive Bass model, suggested by Bretschneider and Mahajan, are used to produce comparative forecasts for a number of different data sets. The approach using the extended Kalman filter is shown to be more robust and almost always more accurate than the alternatives.     

A Practical Method for Designing Linear Quadratic Regulator for Commensurate Fractional-Order Systems

The methods currently available for designing a linear quadratic regulator for fractional-order systems are either based on sufficient-type conditions for the optimality of functionals or generate very complicated analytical solutions even for simple systems. It follows that the use of such methods is limited to very simple problems. The present paper proposes a practical method for designing a linear quadratic regulator (assuming linear state feedback), Kalman filter, and linear quadratic Gaussian regulator/controller for commensurate fractional-order systems (in Caputo sense). For this purpose, considering the fact that in dealing with fractional-order systems the cost function of linear quadratic regulator has only one extremum, the optimal state feedback gains of linear quadratic regulator and the gains of the Kalman filter are calculated using a gradient-based numerical optimization algorithm. Various fractional-order linear quadratic regulator and Kalman filter design problems are solved using the proposed approach. Specifically, a linear quadratic Gaussian controller capable of tracking step command is designed for a commensurate fractional-order system which is non-minimum phase and unstable and has seven (pseudo) states.     

Periodic measurement strategies for distributed parameter systems filtering

This paper deals with the optimization of the output matrix for a discrete time linear stochastic system. The output matrix varies as a periodic function of time, and its values are constrained to belong to a finite prescribed set. The aim is to minimize the average variance of the Kalman filter estimation error in the periodic steady state. The application regards the optimization both of the measurement points and of the scanning sequence for a distributed parameter system (DPS) of parabolic type. A modal approximation is used to reduce the DPS to finite dimension. The proposed solution algorithm makes use of heuristic rules that enable to overcome the difficulties arising from the cardinality of the admissible set, the possible slow convergence of the relevant Riccati equation and the high dimensionality of the lumped approximate model of the DPS. The numerical applications show that the periodic scanning policies, found by the optimization algorithm, cause a great improvement of the filter performance, with respect to the case where a single fixed sensor is used.     

A data assimilation process for linear ill‐posed problems

In this paper, an iteration process is considered to solve linear ill‐posed problems. Based on the randomness of the involved variables, this kind of problems is regarded as simulation problems of the posterior distribution of the unknown variable given the noise data. We construct a new ensemble Kalman filter‐based method to seek the posterior target distribution. Despite the ensemble Kalman filter method having widespread applications, there has been little analysis of its theoretical properties, especially in the field of inverse problems. This paper analyzes the propagation of the error with the iteration step for the proposed algorithm. The theoretical analysis shows that the proposed algorithm is convergence. We compare the numerical effect with the Bayesian inversion approach by two numerical examples: backward heat conduction problem and the first kind of integral equation. The numerical tests show that the proposed algorithm is effective and competitive with the Bayesian method. Copyright © 2017 John Wiley & Sons, Ltd.     

Modelling and nanoscale force spectroscopy of frequency modulation atomic force microscopy

In this paper, a simulation model for frequency modulation atomic force microscopy (FM-AFM) operating in constant amplitude dynamic mode is presented. The model is based on the slow time varying function theory. The mathematical principles to derive the dynamical equations for the amplitude and phase of the FM-AFM cantilever-tip motion are explained and the stability and performance of its closed-loop controller to keep the amplitude at constant value and phase at 90° is analysed. Then, the performance of the theoretical model is supported by comparison of numerical simulations and experiments. Furthermore, the transient behaviour of amplitude, phase and frequency shift of FM-AFM is investigated and the effect of controller gains on the transient motion is analysed. Finally, the derived FM-AFM model is used to simulate the single molecule/nanoscale force spectroscopy and study the effect of sample viscosity, stiffness and Hamaker constant on the response of FM-AFM.     

Online state of charge estimation of Li-ion battery based on an improved unscented Kalman filter approach

An improved unscented Kalman filter approach is proposed to enhance online state of charge estimation in terms of both accuracy and robustness. The goal is to address the drawback associated with the unscented Kalman filter in terms of its requirement for an accurate model and a priori noise statistics. Firstly, Li-ion battery modelling and offline parameter identification is performed. Secondly, a sensitivity analysis experiment is designed to verify which model parameter has the greatest influence on state of charge estimation accuracy, in order to provide an appropriate parameter for the model adaptive algorithm. Thirdly, an improved unscented Kalman filter approach, composed of a model adaptive algorithm and a noise adaptive algorithm, is introduced. Finally, the results are discussed, which reveal that the proposed approach’s estimation error is less than 1.79% with acceptable robustness and time complexity.     

Delay-range-dependent robust H filtering for uncertain stochastic systems with mode-dependent time delays and Markovian jump parameters

This paper investigates the problem of robust H^∞ filtering for uncertain stochastic time-delay systems with Markovian jump parameters. Both the state dynamics and measurement of the system are corrupted by Wiener processes. The time delay varies in an interval and depends on the mode of operation. A Markovian jump linear filter is designed to guarantee robust exponential mean-square stability and a prescribed disturbance attenuation level of the resulting filter error system. A novel approach is employed in showing the robust exponential mean-square stability. The exponential decay rate can be directly estimated using matrices of the Lyapunov-Krasovskii functional and its derivative. A delay-range-dependent condition in the form of LMIs is derived for the solvability of this H^∞ filtering problem, and the desired filter can be constructed with solutions of the LMIs. An illustrative numerical example is provided to demonstrate the effectiveness of the proposed approach.     

An Ensemble Kalman Filter Approach Based on Level Set Parameterization for Acoustic Source Identification Using Multiple Frequency Information

In this paper, a reconstruction problem of the spatial dependent acoustic source from multiple frequency data is discussed. Suppose that the source function is supported on a bounded domain and the piecewise constant intensities of the source are known on the support. We characterize unknown domain by the level set technique. And the level set function can be modeled by a Hamilton-Jacobi system. We use the ensemble Kalman filter approach to analyze the system state. This method can avoid to deal with the nonlinearity directly and reduce the computation complexity. In addition, the algorithm can achieve the stable state quickly with the Hamilton-Jacobi system. From some numerical examples, we show these advantages and verify the feasibility and effectiveness.     

应用DEKF算法训练模糊化神经网络

模糊MLP网络的待定参数规模比确定性MLP网络的多几倍，急需找到高效的训练算法。本文尝试用DEKF训练模糊神经网络，这是DEKF算法新的应用。仿真表明，DEKF算法比经典BP算法的收敛速度更快，训练所得网络的精度更高。     

THE ENSEMBLE KALMAN FILTER FOR MULTIDIMENSIONAL BIOECONOMIC MODELS

To integrate economic considerations into management decisions in ecosystem frameworks, we need to build models that capture observed system dynamics and incorporate existing knowledge of ecosystems, while at the same time accommodating economic analysis. The main constraint for models to serve in economic analysis is dimensionality. In addition, to apply in long‐term management analysis, models should be stable in terms of adjustments to new observations. We use the ensemble Kalman filter to fit relatively simple models to ecosystem or foodweb data and estimate parameters that are stable over the observed variability in the data. The filter also provides a lower bound on the noise terms that a stochastic analysis requires. In this paper, we apply the filter to model the main interactions in the Barents Sea ecosystem. In a comparison, our method outperforms a regression‐based approach.     

The continuous kalman filter as the limit of the discrete kalman filter

In this paper we present a rigorous proof of the commonly held belief that the continuous time Kalman filter equations can be obtained as the limit of the discrete time Kalman filter equations. This is done by creating a uniformly integrable martingale using the discrete filter and showing that its limit, is the continuous filter     

New Approach to Model Validation and Fault Diagnosis

The paper presents a new approach to model validation and fault diagnosis problems for a class of uncertain systems in which the uncertainty is described by an integral quadratic constraint. The new approach is developed by applying methods from linear quadratic optimal control theory. This leads to a method for model validation and fault diagnosis which is based around a robust Kalman filter type structure.     

Pattern recognition based Kalman filter for indoor localization using TDOA algorithm

In this paper, we will present a motion pattern recognition based Kalman filter (PRKF), and apply it to the time difference of arrival (TDOA) algorithm of indoor localization. The state matrix in Kalman filter (KF) is determined by the motion pattern which the target node is supposed to act, and this will bring new system error if the assumption is not correct. Considering this, we first create three fuzzy sets using three KFs whose state matrix stand for different motion patterns, then linearly combined the memberships of a target node of the fuzzy sets. Finally, simulation results show that the PRKF can enhance the localization accuracy about more than 20%.