Multiscale systems, Kalman filters, and Riccati equations

An algorithm analogous to the Rauch-Tung-Striebel algorithm-consisting of a fine-to-coarse Kalman filter-like sweep followed by a coarse-to-fine smoothing step-was developed previously by the authors (ibid. vol.39, no.3, p.464-78 (1994)). In this paper they present a detailed system-theoretic analysis of this filter and of the new scale-recursive Riccati equation associated with it. While this analysis is similar in spirit to that for standard Kalman filters, the structure of the dyadic tree leads to several significant differences. In particular, the structure of the Kalman filter error dynamics leads to the formulation of an ML version of the filtering equation and to a corresponding smoothing algorithm based on triangularizing the Hamiltonian for the smoothing problem. In addition, the notion of stability for dynamics requires some care as do the concepts of reachability and observability. Using these system-theoretic constructs, the stability and steady-state behavior of the fine-to-coarse Kalman filter and its Riccati equation are analysed     

一类动态多尺度系统融合估计算法的分析

为了进一步认识基于状态空间投影的一类动态多尺度系统的融合估计算法本质,本文对该算法进行了分析．首先,将该融合估计算法和在最细尺度上直接进行卡尔曼滤波的算法性能进行了比较,并用仿真进行了验证．结果表明,在最细尺度上,融合估计效果比直接进行卡尔曼滤波的效果好．其次,从计算过程和计算量方面,与一般的时间配准方法进行了对比分析．结果表明,该融合估计算法用比较严谨的数学模型代替了时间配准,可以在每个尺度上获得基于全部观测信息的最优估计,但计算量比时间配准方法大．本文的研究为基于状态空间投影的一类动态多尺度系统的融合估计算法的实际应用奠定了基础．     

Multiscale fuzzy Kalman filtering

Measured data are usually contaminated with errors which sometimes mask their important features. Therefore, data filtering is needed for effective utilization of such measurements. For nonlinear systems which can be described by a Takagi–Sugeno (TS) fuzzy model, several fuzzy Kalman (FK) filtering algorithms have been developed to extend Kalman filtering to such systems. Also, multiscale representation of data is a powerful data analysis tool, which has been successfully used to solve several data filtering problems. In this paper, a multiscale fuzzy Kalman (MSFK) filtering algorithm, in which multiscale representation is utilized to improve the performance of fuzzy Kalman filtering, is developed. The idea is to apply FK filtering at multiple scales to combine the advantages of the FK filter with those of the low pass filters used in multiscale data representation. Starting with a fuzzy model in the time domain, a similar fuzzy model is derived at each scale using the scaled signal approximation of the data obtained by stationary wavelet transform (SWT). These multiscale fuzzy models are then used in FK filtering, and the FK filter with the least cross validation mean square error among all scales is selected as the optimum filter. Also, theoretically, it has been shown that applying FK filtering at a coarser scale than the time domain is equivalent to using a time-averaged FK filter. Finally, the performance of the developed MSFK filtering algorithm is illustrated through a simulated example.     

Linear dynamic filtering with noisy input and output

State estimation problems for linear time-invariant systems with noisy inputs and outputs are considered. An efficient recursive algorithm for the smoothing problem is presented. The equivalence between the optimal filter and an appropriately modified Kalman filter is established. The optimal estimate of the input signal is derived from the optimal state estimate. The result shows that the noisy input/output filtering problem is not fundamentally different from the classical Kalman filtering problem.     

A comparison of simultaneous state and parameter estimation schemes for a continuous fermentor reactor

This article proposes a maximum likelihood algorithm for simultaneous estimation of state and parameter values in nonlinear stochastic state-space models. The proposed algorithm uses a combination of expectation maximization, nonlinear filtering and smoothing algorithms. The algorithm is tested with three popular techniques for filtering namely particle filter (PF), unscented Kalman filter (UKF) and extended Kalman filter (EKF). It is shown that the proposed algorithm when used in conjunction with UKF is computationally more efficient and provides better estimates. An online recursive algorithm based on nonlinear filtering theory is also derived and is shown to perform equally well with UKF and ensemble Kalman filter (EnKF) algorithms. A continuous fermentation reactor is used to illustrate the efficacy of batch and online versions of the proposed algorithms.     

Robust fusion steady‐state filtering for multisensor networked systems with one‐step random delay,missing measurements,and uncertain‐variance multiplicative and additive white noises

The robust fusion steady‐state filtering problem is investigated for a class of multisensor networked systems with mixed uncertainties including multiplicative noises, one‐step random delay, missing measurements, and uncertain noise variances, the phenomena of one‐step random delay and missing measurements occur in a random way, and are described by two Bernoulli distributed random variables with known conditional probabilities. Using a model transformation approach, which consists of augmented approach, derandomization approach, and fictitious noise approach, the original multisensor system under study is converted into a multimodel multisensor system with only uncertain noise variances. According to the minimax robust estimation principle, based on the worst‐case subsystems with conservative upper bounds of uncertain noise variances, the robust local steady‐state Kalman estimators (predictor, filter, and smoother) are presented in a unified framework. Applying the optimal fusion algorithm weighted by matrices, the robust distributed weighted state fusion steady‐state Kalman estimators are derived for the considered system. In addition, by using the proposed model transformation approach, the centralized fusion system is obtained, furthermore the robust centralized fusion steady‐state Kalman estimators are proposed. The robustness of the proposed estimators is proved by using a combination method consisting of augmented noise approach, decomposition approach of nonnegative definite matrix, matrix representation approach of quadratic form, and Lyapunov equation approach, such that for all admissible uncertainties, the actual steady‐state estimation error variances of the estimators are guaranteed to have the corresponding minimal upper bounds. The accuracy relations among the robust local and fused steady‐state Kalman estimators are proved. An example with application to autoregressive signal processing is proposed, which shows that the robust local and fusion signal estimation problems can be solved by the state estimation problems. Simulation example verifies the effectiveness and correctness of the proposed results.     

Optimal and suboptimal smoothing algorithms for identification of time-varying systems with randomly drifting parameters

Noncausal estimation algorithms, which involve smoothing, can be used for off-line identification of nonstationary systems. Since smoothing is based on both past and future data, it offers increased accuracy compared to causal (tracking) estimation schemes, incorporating past data only. It is shown that efficient smoothing variants of the popular exponentially weighted least squares and Kalman filter-based parameter trackers can be obtained by means of backward-time filtering of the estimates yielded by both algorithms. When system parameters drift according to the random walk model and the adaptation gain is sufficiently small, the properly tuned two-stage Kalman filtering/smoothing algorithm, derived in the paper, achieves the Cramér-Rao type lower smoothing bound, i.e. it is the optimal noncausal estimation scheme. Under the same circumstances performance of the modified exponentially weighted least-squares algorithm is often only slightly inferior to that of the Kalman filter-based smoother.     

Edge enhancement by local deconvolution

In this paper a new approach for blurred image restoration is presented. Our algorithm is based on human vision which zooms back and forth in the image in order to identify global structures or details. Deconvolution parameters are estimated by an edge detection and correspond to the ones of a chosen edge detection model. The segmentation is obtained by merging multiscale information provided by multiscale edge detection. The edge detection is achieved by using a derivative approach following a generalization of Canny-Deriche filtering. This multiscale analysis performs an efficient edge detection in noisy blurred images. The merging leads to the best local representation of edge information across scales. The algorithm deals with a mixed (coarse-to-fine/fine-to-coarse) approach and searches for candidate edge points through the scales. Edge characteristics are estimated by the merging algorithm for the chosen model. Scale, direction and amplitude informations allow a local deconvolution of the original image. The noise problem is not considered in this work since it does not disturb the process. Results show that this method allows non-uniformly blurred image restoration. An implementation of the whole algorithm in an intelligent camera (DSP) has been performed.     

多尺度动态模型单传感器动态系统分布式信息融合

利用多尺度分析的思想,将基于模型的动态系统分析方法与基于统计特性的多尺度 信号变换方法相结合,在不同尺度上拥有对目标状态进行不同描述的多模型动态系统,提出 多尺度分布式信息融合估计新算法,在最细尺度上获得目标状态基于全局信息的融合估计 值,初步解决了多尺度动态模型信息融合问题,这些工作丰富和发展了信息融合理论.     

New approach to information fusion steady-state Kalman filtering

By the modern time series analysis method, based on the autoregressive moving average (ARMA) innovation model, a unified and general information fusion steady-state Kalman filtering approach is presented for the general multisensor systems with different local dynamic models and correlated noises. It can handle the filtering, smoothing, and prediction fusion problems for state or signal. The optimal fusion rule weighted by matrices is re-derived as a weighted least squares (WLS) fuser, and is reviewed. An optimal fusion rule weighted by diagonal matrices is presented, which is equivalent to the optimal fusion rule weighted by scalars for components, and it realizes a decoupled fusion. The new algorithms of the steady-state Kalman estimator gains are presented. In order to compute the optimal weights, the formulas of computing the cross-covariances among local estimation errors by Lyapunov equations are presented. The exponential convergence of the iterative solution of Lyapunov equation is proved. It is proved that the optimal fusion estimators under three weighted fusion rules are locally optimal, but are globally suboptimal. The proposed steady-state Kalman fusers can reduce the on-line computational burden, and are suitable for real-time applications. A simulation example for the 3-sensor steady-state Kalman tracking fusion estimators shows their effectiveness and correctness, and gives the accuracy comparison of the fusion rules.     

Nonlinear maximum likelihood estimation of electricity spot prices using recurrent neural networks

Electricity spot prices are complex processes characterized by nonlinearity and extreme volatility. Previous work on nonlinear modeling of electricity spot prices has shown encouraging results, and we build on this area by proposing an Expectation Maximization algorithm for maximum likelihood estimation of recurrent neural networks utilizing the Kalman filter and smoother. This involves inference of both parameters and hyper-parameters of the model which takes into account the model uncertainty and noise in the data. The Expectation Maximization algorithm uses a forward filtering and backward smoothing (Expectation) step, followed by a hyper-parameter estimation (Maximization) step. The model is validated across two data sets of different power exchanges. It is found that after learning a posteriori hyper-parameters, the proposed algorithm outperforms the real-time recurrent learning and the extended Kalman Filtering algorithm for recurrent networks, as well as other contemporary models that have been previously applied to the modeling of electricity spot prices.     

两轮自平衡机器人惯性传感器滤波问题的研究

针对惯性传感器在两轮机器人姿态检测中存在随机漂移误差的问题,基于卡尔曼滤波实现对倾角仪与陀螺仪的信息融合,设计了简单而实用的滤波算法,对传感器的误差进行补偿后得到机器人姿态信号的最优估计,从而将其应用于两轮自平衡机器人系统。实验结果表明,采用卡尔曼信息融合的方法,来得到机器人姿态信息最优估计是有效可行的,并且有利于机器人完成自平衡的控制。     

Multiscale smoothing error models

A class of multiscale stochastic models based on scale-recursive dynamics on trees has recently been introduced. These models are interesting because they can be used to represent a broad class of physical phenomena and because they lead to efficient algorithms for estimation and likelihood calculation. In this paper, we provide a complete statistical characterization of the error associated with smoothed estimates of the multiscale stochastic processes described by these models. In particular, we show that the smoothing error is itself a multiscale stochastic process with parameters that can be explicitly calculated     

基于补偿卡尔曼滤波的姿态估计算法实现

九轴的姿态测量单元包括加速度计、磁力计和陀螺仪,它采用微机电加工工艺,价格低廉但精度较差.为了获得更好的滤波稳定性和估计精度,基于补偿卡尔曼滤波原理,讨论了姿态估计算法的实现.具体过程如下:基于补偿卡尔曼滤波原理,讨论九轴测量信息的融合方法;利用加速度计和磁力计估算姿态初始值;使用手动转台实现静态标定,纠正零偏;使用高精度陀螺仪STIM210实现动态标定,设置滤波系数.结果表明,估计算法具有良好性能.     

Adaptive Kalman Filtering and Smoothing for Tracking Vocal Tract Resonances Using a Continuous-Valued Hidden Dynamic Model

A novel Kalman filtering/smoothing algorithm is presented for efficient and accurate estimation of vocal tract resonances or formants, which are natural frequencies and bandwidths of the resonator from larynx to lips, in fluent speech. The algorithm uses a hidden dynamic model, with a state-space formulation, where the resonance frequency and bandwidth values are treated as continuous-valued hidden state variables. The observation equation of the model is constructed by an analytical predictive function from the resonance frequencies and bandwidths to LPC cepstra as the observation vectors. This nonlinear function is adaptively linearized, and a residual or bias term, which is adaptively trained, is added to the nonlinear function to represent the iteratively reduced piecewise linear approximation error. Details of the piecewise linearization design process are described. An iterative tracking algorithm is presented, which embeds both the adaptive residual training and piecewise linearization design in the Kalman filtering/smoothing framework. Experiments on estimating resonances in Switchboard speech data show accurate estimation results. In particular, the effectiveness of the adaptive residual training is demonstrated. Our approach provides a solution to the traditional "hidden formant problem," and produces meaningful results even during consonantal closures when the supra-laryngeal source may cause no spectral prominences in speech acoustics     

Regularization networks: fast weight calculation via Kalman filtering.

Regularization networks are nonparametric estimators obtained from the application of Tychonov regularization or Bayes estimation to the hypersurface reconstruction problem. Their main drawback back is that the computation of the weights scales as O(n(3)) where n is the number of data. In this paper, we show that for a class of monodimensional problems, the complexity can be reduced to O(n) by a suitable algorithm based on spectral factorization and Kalman filtering. Moreover, the procedure applies also to smoothing splines.     

广义系统Kalman融合器的研究

研究带自回归滑动平均(ARMA)有色观测噪声的多传感器广义离散随机线性系统,根据Kalman滤波方法和白噪声估计理论,在线性最小方差信息融合准则下,应用奇异值分解和增广状态空间模型,为了提高融合器的精度,提出了按矩阵加权降阶稳态广义Kalman融合器,可统一处理稳态滤波、平滑和预报问题,可减少计算负担和改善局部估计精度。并提出最优加权系数的局部估计误差方差和协方差阵的计算公式。用一个Monte Carlo数值仿真实例说明了所提方法的有效性。     

传感器网络中信息融合的几个基本问题

系统地阐述了传感器网络环境中几个基本而又重要的信息融合问题的最近进展，包括：最一般条件下全局最优的多传感器分布式统计判决；传感器观测数据或局部估计的最优维数压缩；一般条件下最优线性无偏估计融合公式及其有效算法；传感器观测噪声相关情形下动态系统的卡尔曼滤波融合；容错条件下的区间估计融合．这些结果对传感器网络的设计与应用具有重要意义．     

非线性离散系统的相关观测融合时变Kalman滤波

针对带相关观测噪声和带不同观测函数的多传感器离散非线性系统,利用推广的离散Kalman滤波方法对状态系统和观测系统进行线性化处理,提出了基于岭估计的加权最小二乘(REWLS)分布式融合Kalman滤波算法.以风险函数为评价指标,利用信息滤波器比较了各种观测融合Kalman滤波算法,其中REWLS分布式融合算法精度最高.同时,分布式融合算法减少了计算负担,便于实时应用.仿真例子表明了理论分析的正确性.     

基于Kalman滤波的自回归滑动平均信号信息融合Wiener滤波器

应用Kalman滤波方法,在按矩阵加权线性最小方差最优信息融合规则下,提出了带白色观测噪声的多通道ARMA信号的多传感器信息融合Wiener滤波器.它可统一处理信息融合滤波、平滑和预报问题.为了计算最优加权阵,提出了计算局部滤波误差互协方差阵的公式.同单传感器情形相比,可提高估计精度.一个带三传感器的目标跟踪系统的仿真例子说明了其有效性.