Finite-dimensional recurrent algorithms for optimal nonlinear logical–dynamical filtering

The problem of the most accurate estimation of the current state of a multimode nonlinear dynamic observation system with discrete time based on indirect measurements of this state is considered. The general case when a mode indicator is available and the measurement errors depend on the plant disturbances is investigated. A comparative analysis of two known approaches is performed—the conventional absolutely optimal one based on the use of the posterior probability distribution, which requires the use of an unimplementable infinite-dimensional estimation algorithm, and a finitedimensional optimal approach, which produces the best structure of the difference equation of a low-order filter. More practical equations for the Gaussian approximations of these two optimal filters are obtained and compared. In the case of the absolutely optimal case, such an approximation is finitedimensional, but it differs from the approximation of the finite-dimensional optimal version in terms of its considerably larger dimension and the absence of parameters. The presence of parameters, which can be preliminarily calculated using the Monte-Carlo method, allows the Gaussian finite-dimensional optimal filter to produce more accurate estimates.     

Optimal continuous-discrete nonlinear finite memory filter with a discrete predictions

We consider the following problem: using the discrete time measurements of the state variables of a continuous stochastic object and obtain the most precise estimate of these variables. To accelerate the estimating process, we synthesize the optimal structure of a discrete final-dimensional nonlinear filter with a piecewise-constant prediction, remembering the last few measurements in its state vector. The dimension of this vector (i.e., the filter memory volume) can be selected arbitrarily to balance the desired measurement precision and available processing speed for the measurements. We obtain the mean-square optimal structure functions of the filter expressed via the corresponding probability distributions and a chain of Fokker—Planck—Kolmogorov equations to find those distributions. We describe the procedure to obtain the structural functions of the filter numerically by means of the Monte-Carlo method. Also, we provide simple numeric-analytic approximations to the proposed filters: they are compared with approximations to the known filters. An example of the construction of such approximations is considered.     

A optimal discrete nonlinear arbitrary-order filter

It is proposed to synthesize a recurrent nonlinear filter of any given order chosen from the condition of obtaining online estimates in the course of measurements for estimation of a Markov random sequence. Optimality conditions for structural functions of the filter are obtained. The application of the gradient descent method is considered. Relations of this filter with the Stratonovich filter of an infinite order and a filter of optimal structure of an object order is established. Gaussian and linearized approximations to the optimal filter of an arbitrary order are constructed. An example is presented.     

高斯渐进贝叶斯滤波器

渐进贝叶斯方法将先验分布到后验分布的演化描述为一阶动态系统,通过在伪时间上连续地引入观测信息实现后验状态估计.该方法的一般形式解,即动态系统的时间导数,是难以得到的.本文提出一种高斯型渐进贝叶斯滤波器.首先在线性高斯条件下推导了时间导数的解析解;然后证明了在该条件下,由该解析解确定的一阶动态系统与常量状态估计的Kalman-Bucy滤波器是一致的,且由此导出的高斯渐进贝叶斯滤波器与卡尔曼滤波器是一致的.最后利用一阶Taylor展开推导了滤波器在非线性高斯条件下的近似解表达式,并采用Monte Carlo方法给出了具体实现方法.通过若干仿真算例表明,新滤波器具有较高的精度,且在一定精度条件下的时间复杂度低于一般粒子滤波器.     

Performance assessment,diagnosis, and optimal selection of non-linear state filters

Non-linear state filters of different approximations and capabilities allow for real-time estimation of unmeasured states in non-linear stochastic processes. It is well known that the performance of non-linear filters depends on the underlying numerical and statistical approximations used in their design. Despite the theoretical and practical interest in evaluating the performance of non-linear filtering methods, it remains one of the most complex problems in the area of state estimation. We propose the use of posterior Cramér–Rao lower bound (PCRLB) or mean square error (MSE) inequality as a filtering performance benchmark. Using the PCRLB inequality, we develop assessment and diagnosis tools for monitoring and evaluating the performance of non-linear filters. Using the PCRLB inequality-based performance assessment tool, an optimal non-linear filter switching strategy is proposed for state estimation in general non-linear systems. The non-linear filter switching strategy is an optimal performance strategy, which maintains high filtering performance under all operating conditions. The complex, high dimensional integrals involved in the computation of the PCRLB inequality-based non-linear filter assessment and diagnosis tools are approximated using sequential Monte-Carlo (SMC) methods. The utility and efficacy of the developed tools are illustrated through a numerical example.     

基于条件高斯分布的未知输入与状态估计算法

针对未知输入同时存在于系统方程和测量方程的直接馈通线性随机系统, 提出了一种同时估计未知输入 和状态的算法. 首先, 通过将未知输入模型描述为有限方差的高斯分布, 利用条件高斯分布的性质, 推导出新的滤波 算法, 以同时得到未知输入估计和状态估计. 其次, 证明了当未知输入的方差趋于无穷大时, 本文提出的算法等价于 已有的递归三步滤波算法. 最后, 分析了本文算法的渐进稳定性条件, 结果表明, 与已有算法相比, 本文的算法适用 范围更广.     

Uncertainties for recursive estimators in nonlinear state-space models, with applications to epidemiology

Consider a nonlinear dynamic system where one wishes to estimate a state vector using noisy measurements. Many algorithms have been proposed to address this problem, among them the extended Kalman filter (and its variants) and constant-gain stochastic approximation. To quantify the efficacy of these algorithms, it is necessary to describe the distribution of the state estimation error. Typically, performance has been measured by the estimation error covariance alone, which does not provide enough information to probabilistically quantify the estimation accuracy. By casting the estimation error in an autoregressive-type form, this paper addresses the broader question of the distribution of the error for a general class of recursive algorithms. We illustrate the distributional results in an epidemiological problem of disease monitoring.     

基于中心差分粒子滤波的SLAM算法

针对移动机器人同时定位与地图创建(Simultaneous localization and mapping, SLAM)中的FastSLAM算法, 存在非线性系统线性化处理和计算雅可比矩阵的缺点, 本文提出了基于Sterling多项式插值处理非线性系统的SLAM方法. 该方法基于Rao-Blackwellized粒子滤波框架, 利用中心差分滤波方法产生改进的建议分布函数, 提高了机器人位姿估计的精度; 利用中心差分滤波初始化特征和更新地图中的特征, 提高了地图创建的精度; 针对实际应用中存在虚假特征的情况 提出了一种有效的地图管理方法. 在同等粒子数的情况下, 该方法改进了SLAM结果的精度. 基于仿真和实际数据的实验结果验证了该方法的有效性.     

Analytical-numerical approximations of the optimal recurrent logical—dynamical low order filter-predictor

Practical methods for constructing suboptimal approximations of the most accurate fast two-step finite-dimensional nonlinear estimator of the current random operation mode and state vector of a multi-mode discrete time plant based on indirect measurements are considered. The proposed methods are based on Gaussian approximations of certain probability densities and on a possible linearization of the plant’s and measuring instrument’s nonlinearities if they are sufficiently smooth. As a result, the suboptimal structure functions of the finite-dimensional filter-predictor are analytically expressed either in terms of the characteristics of the statistical linearization of these nonlinearities or in terms of these nonlinearities themselves and their first-order partial derivatives. The parameters of these functions are determined numerically by finding the probabilities, expectations, and covariances using the Monte Carlo method. The proposed approach is compared with its well-known absolutely optimal analog of a much higher order and with a similar approximation to a one-step finite-dimensional low order filter.

     

Estimating Vehicle State by GPS/IMU Fusion with Vehicle Dynamics

In this paper, a low-cost navigation system with high integrity and reliability is proposed. A high-integrity estimation filter is proposed to obtain a high-accuracy state estimate. The filter utilizes a vehicle velocity constraint measurement to enhance the accuracy of the estimate. Two estimation filters, the extended Kalman filter (EKF) and the extended information filter (EIF), are designed and compared to obtain the estimate of the vehicle state. An instrumentation system that consists of a microcontroller, GPS receiver, IMU, velocity encoder, and Zigbee transceiver is used. The microcontroller provides a vehicle navigation solution at 50 Hz by fusing the measurements of the IMU and GPS receiver using the proposed filter design. Extensive experimental tests are conducted to verify the accuracy of the proposed algorithm. These results are processed with and without the velocity constraints. The estimation accuracy improvement with the addition of the velocity constraints is shown. A more than 16 % reduction in the computational time is demonstrated when using the EIF in comparison to the EKF approach.     

带随机量测时滞和随机丢包的改进无迹卡尔曼滤波

经典卡尔曼滤波要求量测值可实时获取,且仅适用于线性系统.然而,在工程实际应用中,系统多为非线性系统,量测值也会发生滞后或者丢失等现象,此时经典卡尔曼滤波已不适用.因此,本文针对一类带有随机量测一步时滞和随机丢包的非线性离散系统的状态估计问题,用两个满足伯努利分布的独立随机变量来描述随机量测一步滞后和随机丢包的现象.当量测丢失时,用量测值的一步预测值来代替零输入进行补偿.在此基础上应用正交投影理论和无迹变换的方法提出了一种改进的无迹卡尔曼滤波算法.最后,通过仿真例子验证在考虑随机量测一步时滞和随机丢包的情况下,所提出的改进算法相比于经典无迹卡尔曼滤波算法具有更高的精度.     

带约束卡尔曼滤波对涡扇发动机状态估计

提出了一种加入线性不等式约束的卡尔曼滤波方法,并用于涡扇发动机的健康状况估计。涡扇发动机数字模型包含10个状态变量、12个量测量、6个控制输入量以及8个健康状况参数。不等式约束不仅保证了状态变量估计在用户自定义的范围内随时间变化平稳缓慢,而且还提高了滤波计算效率,改善了滤波估计精度。同时系统还允许滤波器沿确定的方向修正状态变量估计,以保持状态变量真值恒定。对比传统的无约束卡尔曼滤波,线性化滤波结果显示,该方法对涡扇发动机的健康状况估计尤其行之有效。     

Least squares estimation of nonstationary covariance parameters in linear systems

Least squares estimation techniques are employed to overcome previous difficulties encountered in accurately estimating the state and measurement noise covariance parameters in linear stochastic systems. In the past accurate and rapidly converging covariance parameter estimates have been achieved with complex estimation algorithms only after specifying the statistical nature of the noise in the system and constraining the time variation of the covariance parameters. Weighted least squares estimation allows these restrictions to be removed while achieving near optimal accuracy using a filter on the same order of complexity as a Kalman filter. Allowing the covariance parameters to vary in as general a manner in time as the state in a linear discrete time stochastic system, and assuming that a Kalman filter is applied to this system using incorrect knowledge of the a priori statistics, it is shown how a covariance system is developed similar to the original system. Unbiased least squares estimates of the covariance parameters and of the original state are obtained without the necessity of specifying the distribution on the noise in either system. The accuracy of these estimates approaches optimal accuracy with increasing measurements when adaptive Kalman filters are applied to each system.     

A Novel Fuzzy Approach for State Estimation of Nonlinear Hybrid Systems Using Particle Filtering Method

Reliable state estimation is challenging for nonlinear hybrid systems. Particle filtering has emerged as an appealing approach for online hybrid state estimation. Mode detection in nonlinear hybrid systems is, however, a troublesome issue for the conventional particle filter mainly due to sample impoverishment. The problem is also exacerbated when dynamics that govern healthy or faulty modes are close together. False mode detection consequently leads to erroneous continuous state estimation. This paper proposes a novel fuzzy‐based particle filter to reduce continuous state estimation errors due to failures in mode detection. It is fulfilled by considering a fuzzified contribution of each feasible mode in overall estimation. In addition, two new resampling strategies are presented to tackle the degeneracy problem. A set of simulation test studies are conducted to extract the characteristic features and evaluate the performance of the proposed algorithm compared to observation and transition‐based most likely modes tracking particle filter (OTPF) as one of the most meticulous proposed estimation algorithms. The simulation results demonstrate the superior efficiency of the algorithm in dealing with the considered potential estimation problems.     

以改善精度为目标的人手跟踪方法研究

分别从UKF滤波器的内在机理和人手运动模型两个方面入手,以改善跟踪结果的精确度为基本目标,重点对UKF算法中存在的部分理论问题进行了探讨,在此基础上提出了改进后的UKFDUT算法,同时也对IMM进行了改进,把IMM模型变为MM模型,再进一步将UKFDUT算法和MM模型相融合,得到UKFDUT MM算法,研究表明,Sigma点具有一些特性,通过对这些特性进行研究,可以找到改进跟踪精度的新途径;把MM模型和人手模型评价标准相结合,可以取得比单独使用IMM更好的跟踪精度,实验结果也表明了算法的有效性和令人满意的跟踪精度.     

基于多元插值的DD型滤波算法仿真研究

DD型滤波是一种基于Striling多元插值方法,将函数按多项式近似展开的非线性滤波算法.相对于扩展卡尔曼滤波而言,它不需要对非线性函数进行微分运算,具有滤波精度高、数值稳定性好和适用范围广的优点,其运算量却与扩展卡尔曼滤波相当.对DD型滤波算法进行了深入分析,并将该算法应用于状态估计领域.对一多传感器目标跟踪问题进行了仿真计算,仿真结果表明了DD型滤波算法的有效性和实用性.     

Recursive state estimation for discrete‐time nonlinear systems with event‐triggered data transmission,norm‐bounded uncertainties and multiple missing measurements

In this paper, we consider the recursive state estimation problem for a class of discrete‐time nonlinear systems with event‐triggered data transmission, norm‐bounded uncertainties, and multiple missing measurements. The phenomenon of event‐triggered communication mechanism occurs only when the specified event‐triggering condition is violated, which leads to a reduction in the number of excessive signal transmissions in a network. A sequence of independent Bernoulli random variables is employed to model the multiple measurements missing in the transmission. The norm‐bounded uncertainties that could be considered as external disturbances which lie in a bounded set. The purpose of the addressed filtering problem is to obtain an optimal robust recursive filter in the minimum‐variance sense such that with the simultaneous presence of event‐triggered data transmission, norm‐bounded uncertainties, and multiple missing measurements; the filtering error is minimized at each sampling time. By solving two Riccati‐like difference equations, the filter gain is calculated recursively. Based on the stochastic analysis theory, it is proved that the estimation error is bounded under certain conditions. Finally, two numerical examples are presented to demonstrate the effectiveness of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

Neural networks for nonlinear state estimation

Estimating the state of a nonlinear stochastic system (observed through a nonlinear noisy measurement channel) has been the goal of considerable research to solve both filtering and control problems. In this paper, an original approach to the solution of the optimal state estimation problem by means of neural networks is proposed, which consists in constraining the state estimator to take on the structure of a multilayer feedforward network. Both non-recursive and recursive estimation schemes are considered, which enable one to reduce the original functional problem to a nonlinear programming one. As this reduction entails approximations for the optimal estimation strategy, quantitative results on the accuracy of such approximations are reported. Simulation results confirm the effectiveness of the proposed method.     

State and parameter estimation for nonlinear biological phenomena modeled by S-systems

Biological pathways can be modeled as a nonlinear system described by a set of nonlinear ordinary differential equations (ODEs). A central challenge in computational modeling of biological systems is the determination of the model parameters. In such cases, estimating these variables or parameters from other easily obtained measurements can be extremely useful. For example, time-series dynamic genomic data can be used to develop models representing dynamic genetic regulatory networks, which can be used to design intervention strategies to cure major diseases and to better understand the behavior of biological systems. Unfortunately, biological measurements are usually highly affected by errors that hide the important characteristics in the data. Therefore, these noisy measurements need to be filtered to enhance their usefulness in practice. This paper addresses the problem of state and parameter estimation of biological phenomena modeled by S-systems using Bayesian approaches, where the nonlinear observed system is assumed to progress according to a probabilistic state space model. The performances of various conventional and state-of-the-art state estimation techniques are compared. These techniques include the extended Kalman filter (EKF), unscented Kalman filter (UKF), particle filter (PF), and the developed improved particle filter (IPF). Specifically, two comparative studies are performed. In the first comparative study, the state variables (the enzyme CadA, the transport protein CadB, the regulatory protein CadC and lysine Lys for a model of the Cad System in E. coli (CSEC)) are estimated from noisy measurements of these variables, and the various estimation techniques are compared by computing the estimation root mean square error (RMSE) with respect to the noise-free data. In the second comparative study, the state variables as well as the model parameters are simultaneously estimated. In this case, in addition to comparing the performances of the various state estimation techniques, the effect of the number of estimated model parameters on the accuracy and convergence of these techniques is also assessed. The results of both comparative studies show that the UKF provides a higher accuracy than the EKF due to the limited ability of EKF to accurately estimate the mean and covariance matrix of the estimated states through lineralization of the nonlinear process model. The results also show that the IPF provides a significant improvement over PF because, unlike the PF which depends on the choice of sampling distribution used to estimate the posterior distribution, the IPF yields an optimum choice of the sampling distribution, which also accounts for the observed data. The results of the second comparative study show that, for all techniques, estimating more model parameters affects the estimation accuracy as well as the convergence of the estimated states and parameters. However, the IPF can still provide both convergence as well as accuracy related advantages over other estimation methods.     

带未知通信干扰和丢包补偿的多传感器网络化不确定系统的分布式融合滤波

研究了带有未知通信干扰、观测丢失和乘性噪声不确定性的多传感器网络化系统的状态估计问题.通过白色乘性噪声描述系统状态和观测中的随机不确定性,采用一组服从Bernoulli分布的随机变量描述网络传输过程中存在的观测丢失现象,且数据传输中存在未知的网络通信干扰.当发生丢包时,以当前丢失观测的预报值进行补偿.对每个单传感器子系统,应用线性无偏最小方差估计准则设计了不依赖于未知通信干扰的最优线性滤波器.推导了任两个局部滤波误差之间的互协方差阵.进而,应用矩阵加权融合估计算法给出了分布式融合状态滤波器.仿真例子验证了算法的有效性.