Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants

This work studies the effects of sampling variability in Monte Carlo-based methods to estimate very high-dimensional systems. Recent focus in the geosciences has been on representing the atmospheric state using a probability density function, and, for extremely high-dimensional systems, various sample-based Kalman filter techniques have been developed to address the problem of real-time assimilation of system information and observations. As the employed sample sizes are typically several orders of magnitude smaller than the system dimension, such sampling techniques inevitably induce considerable variability into the state estimate, primarily through prior and posterior sample covariance matrices. In this article, we quantify this variability with mean squared error measures for two Monte Carlo-based Kalman filter variants: the ensemble Kalman filter and the ensemble square-root Kalman filter. Expressions of the error measures are derived under weak assumptions and show that sample sizes need to grow proportionally to the square of the system dimension for bounded error growth. To reduce necessary ensemble size requirements and to address rank-deficient sample covariances, covariance-shrinking (tapering) based on the Schur product of the prior sample covariance and a positive definite function is demonstrated to be a simple, computationally feasible, and very effective technique. Rules for obtaining optimal taper functions for both stationary as well as non-stationary covariances are given, and optimal taper lengths are given in terms of the ensemble size and practical range of the forecast covariance. Results are also presented for optimal covariance inflation. The theory is verified and illustrated with extensive simulations.     

Discrete-time reduced-order estimator design for bilinear stochastic systems with error covariance assignment

This paper is concerned with the design of reduced-order stateestimators for bilinear stochastic discrete-time systems subjectedto estimation error covariance assignment. The purpose of theproblem addressed is to design the reduced-order state estimators for the bilinear stochastic discrete-time systems such thatthe steady-state estimation error covariances achieve the prespecified values. A simple, effective matrix inequality approach is developed to solve this problem. Specifically, (1) the parameterisationof estimation error covariances that certain bilinear errordynamic processes may possess is presented, (2) the characterisationof all reduced-order state estimators that assign such errorcovariances is explicitly derived, and (3) the solvabilityof the assignability conditions is discussed. Furthermore,an illustrative example is used to demonstrate the effectivenessof the proposed design procedure.     

The optimal robust finite-horizon Kalman filtering for multiple sensors with different stochastic failure rates

This work deals with the filtering problem for norm-bounded uncertain discrete dynamic systems with multiple sensors having different stochastic failure rates. For tackling the uncertainties of the covariance matrices of state and state estimation error simultaneously, their upper bounds containing a scaling parameter are derived, and then a robust finite-horizon filtering minimizing the upper bound of the estimation error covariance is proposed. Furthermore, an optimal scaling parameter is exploited to reduce the conservativeness of the upper bounds of the state and estimation error covariances, which leads to an optimal robust filtering for all possible missing measurements and all admissible parameter uncertainties. A numerical example illustrates the performance improvement over the traditional Kalman filtering method.     

Estimation of systems with statistically-constrained inputs

This paper discusses the estimation of a class of discrete-time linear stochastic systems with statistically-constrained unknown inputs (UI), which can represent an arbitrary combination of a class of un-modeled dynamics, random UI with unknown covariance matrix and deterministic UI. In filter design, an upper bound filter is explored to compute, recursively and adaptively, the upper bounds of covariance matrices of the state prediction error, innovation and state estimate error. Furthermore, the minimum upper bound filter (MUBF) is obtained via online scalar parameter convex optimization in pursuit of the minimum upper bounds. Two examples, a system with multiple piecewise UIs and a continuous stirred tank reactor (CSTR), are used to illustrate the proposed MUBF scheme and verify its performance.     

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.     

A Kalman filter as a minimax estimator

A minimax terminal state estimation problem is posed for a linear plant and a generalized quadratic loss function. Sufficient conditions are developed to insure that a Kalman filter will provide a minimax estimate for the terminal state of the plant. It is further shown that this Kalman filter will not generally be a minimax estimate for the terminal state if the observation interval is arbitrarily long. Consequently, a subminimax estimate is defined, subject to a particular existence condition. This subminimax estimate is related to the Kalman filter, and it may provide a useful estimate for the terminal state when the performance of the Kalman filter is no longer satisfactory.     

Mobile Robot Global Localization using an Evolutionary MAP Filter

A new algorithm based on evolutionary computation concepts is presented in this paper. This algorithm is a non linear evolutive filter known as the Evolutive Localization Filter (ELF) which is able to solve the global localization problem in a robust and efficient way. The proposed algorithm searches stochastically along the state space for the best robot pose estimate. The set of pose solutions (the population) represents the most likely areas according to the perception and motion information up to date. The population evolves by using the log-likelihood of each candidate pose according to the observation and the motion error derived from the comparison between observed and predicted data obtained from the probabilistic perception and motion model. The algorithm has been tested on a mobile robot equipped with a laser range finder to demonstrate the effectiveness, robustness and computational efficiency of the proposed approach.     

在不确定观测下离散状态时滞系统的最优滤波

研究了在不确定观测下离散状态时滞系统的最优滤波问题,观测值的不确定性则通过一个满足Bernoulli分布且统计特性已知的随机变量来描述. 一般采用状态增广方法将时滞系统转换为无时滞随机系统, 再利用Kalman滤波器的设计方法解决最优状态估计问题, 但是当系统时滞较大时,转换后的系统状态维数很高, 这样增加了计算负担. 为此,基于最小方差估计准则, 利用射影性质和递归射影公式得到了一个新的滤波器设计方法, 而且保证了滤波器的维数与原系统相同.最后, 给出一个仿真例子说明所提方法的有效性.     

An alternative technique for solving a coupled PDE system in interphase heat transfer

Recently an accelerated iterative procedure was studied for solving a coupled partial differential equation system in interphase heat transfer to improve some existing iterative procedures in the literature. In that procedure, at each step of the iteration one has to evaluate the derivative of a well-known function at a new point. In this paper, an alternative approach is proposed in which one has to evaluate the derivative only once throughout the procedure. The proposed new iterative scheme also has the same order of convergence and takes lesser number of iterations for certain benchmark problems. An interesting theoretical study on the monotone convergence as well as error estimate of the proposed iterative procedure are provided for continuous as well as discretized problems. The proposed iterative procedure also supplements the existence and uniqueness of the solution in both the cases. A comparative numerical study is also done to demonstrate the efficacy of the proposed scheme.     

Linear Kalman–Bucy Filter with Autoregressive Signal and Noise

In the Kalman—Bucy filter problem, the observed process consists of the sum of a signal and a noise. The filtration begins at the same moment as the observation process and it is necessary to estimate the signal. As a rule, this problem is studied for the scalar and vector Markovian processes. In this paper, the scalar linear problem is considered for the system in which the signal and noise are not Markovian processes. The signal and noise are independent stationary autoregressive processes with orders of magnitude higher than 1. The recurrent equations for the filter process, its error, and its conditional cross correlations are derived. These recurrent equations use previously found estimates and some last observed data. The optimal definition of the initial data is proposed. The algebraic equations for the limit values of the filter error (the variance) and cross correlations are found. The roots of these equations make possible the conclusions concerning the criterion of the filter process convergence. Some examples in which the filter process converges and does not converge are given. The Monte Carlo method is used to control the theoretical formulas for the filter and its error.