Optimal filtering for linear systems with state and multiple observation delays

This article solves the optimal filtering problem for linear systems with state and multiple observation delays. The optimal estimate equation similar to the traditional Kalman–Bucy one is derived, and the system of equations for determining the filter gain matrix consists of an infinite set of equations. It is then demonstrated that a finite set of the filtering equations can be obtained in case of commensurable delays. In the example, the designed optimal filter is compared to the traditional Kalman–Bucy filter.     

Optimal linear filtering over observations with multiple delays

In this paper, the optimal filtering problem for a linear system over observations with multiple delays is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and its variance. As a result, the optimal filtering equations similar to the traditional Kalman–Bucy ones are obtained in the form dual to the Smith predictor, commonly used for robust control design in time‐delay systems. In the example, the obtained optimal filter over observations with multiple delays is verified for a sample system and compared with the best Kalman–Bucy filter available for delayed measurements. Copyright © 2004 John Wiley & Sons, Ltd.     

Optimal filtering for linear state delay systems

In this note, the optimal filtering problem for linear systems with state delay over linear observations is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. As a result, the optimal estimate equation similar to the traditional Kalman-Bucy one is derived; however, it is impossible to obtain a system of the filtering equations, that is closed with respect to the only two variables, the optimal estimate and the error variance, as in the Kalman-Bucy filter. The resulting system of equations for determining the error variance consists of a set of equations, whose number is specified by the ratio between the current filtering horizon and the delay value in the state equation and increases as the filtering horizon tends to infinity. In the example, performance of the designed optimal filter for linear systems with state delay is verified against the best Kalman-Bucy filter available for linear systems without delays and two versions of the extended Kalman-Bucy filter for time-delay systems.     

Optimal filtering for polynomial system states with polynomial multiplicative noise

In this paper, the optimal filtering problem for polynomial system states with polynomial multiplicative noise over linear observations is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. As a result, the Ito differentials for the optimal estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining a closed system of the filtering equations for any polynomial state with polynomial multiplicative noise over linear observations is then established, which yields the explicit closed form of the filtering equations in the particular cases of a linear state equation with linear multiplicative noise and a bilinear state equation with bilinear multiplicative noise. In the example, performance of the designed optimal filter is verified for a quadratic state with a quadratic multiplicative noise over linear observations against the optimal filter for a quadratic state with a state‐independent noise and a conventional extended Kalman–Bucy filter. Copyright © 2006 John Wiley & Sons, Ltd.     

Application of optimal filtering methods for on-line of queueing network states

We show the solution to the optimal filtering problem for states of Markov jump processes by observations of multivariant point processes. A characteristic feature of observations is that their compensators are random linear functions of the system state, and the composite “state–observations” process does not possess the Markov property. The provided optimal filtering estimate is expressed via the solution of some recurrent system of linear differential equations and algebraic relations. We present examples of using theoretical results to construct typical models of real queueing networks. We establish the connections between our new optimal filtering algorithm and classical results of Kalman–Bucy and Wonham. We propose a solution for the problem of estimating the current state of a UDP connection given the observations of video stream.     

Solution in the z-domain to the discrete-time stationary Kalman filtering problem for systems with perfect measurements

The discrete-time stationary Kalman filtering problem is solved in thez-domain for general right invertible linear systems whose measurements are all noise free. A simple expression in closed form is obtained for the transfer function of the optimal filter both for the uniform and the general nonuniform rank cases. It provides an optimal solution also for systems with uniform and nonuniform time delays. The minimum variance error of the estimate is analyzed and the independent contributions to this error of the system zeros outside the unit circle, finite and infinite, and the excess of the number of inputs over the number of measurements are investigated.     

Optimal state estimation using randomly delayed measurements without time stamping

This paper studies an optimal state estimation (Kalman filtering) problem under the assumption that output measurements are subject to random time delays caused by network transmissions without time stamping. We first propose a random time delay model which mimics many practical digital network systems. We then study the so‐called unbiased, uniformly bounded linear state estimators and show that the estimator structure is given based on the average of all received measurements at each time for different maximum time delays. The estimator gains can be derived by solving a set of recursive discrete‐time Riccati equations. The estimator is guaranteed to be optimal in the sense that it is unbiased with uniformly bounded estimation error covariance. A simulation example shows the effectiveness of the proposed algorithm. Copyright © 2013 John Wiley & Sons, Ltd.     

不确定离散系统的最优鲁棒滤波

本文对一类含有范数有界参数不确定的离散线性系统的滤波问题进行了研究,了有限时域时变以及无限时域时不变两种情形,给出了一个对所有可容许参数不确定都能满足的估计误差方差上界,得到了使得该上界达到最小的最优鲁棒滤波器形式及其存在的充要条件,数值结果表明：当系统存在参数不确定时,本文所得到的滤波器优于标准的Ｋａｌｍａｎ滤波器以及文（４）中的鲁棒滤波器。     

Optimal distributed Kalman filtering fusion for a linear dynamic system with cross-correlated noises

In this article, we study the distributed Kalman filtering fusion problem for a linear dynamic system with multiple sensors and cross-correlated noises. For the assumed linear dynamic system, based on the newly constructed measurements whose measurement noises are uncorrelated, we derive a distributed Kalman filtering fusion algorithm without feedback, and prove that it is an optimal distributed Kalman filtering fusion algorithm. Then, for the same linear dynamic system, also based on the newly constructed measurements, a distributed Kalman filtering fusion algorithm with feedback is proposed. A rigorous performance analysis is dedicated to the distributed fusion algorithm with feedback, which shows that the distributed fusion algorithm with feedback is also an optimal distributed Kalman filtering fusion algorithm; the P matrices are still the estimate error covariance matrices for local filters; the feedback does reduce the estimate error covariance of each local filter. Simulation results are provided to demonstrate the validity of the newly proposed fusion algorithms and the performance analysis.     

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.     

An application of the information theory to filtering problems

The purpose of this paper is to study the filtering problems from the viewpoint of the information theory. For a linear system it is proved that the necessary and sufficient condition for maximizing the mutual information between a state and the estimate is to minimize the entropy of the estimation error. Then we derive the Kalman-Bucy filter for both the discrete-time and the continuous-time systems by an application of the information theory. Furthermore, the approach is extended to the nonlinear dynamical systems with noisy observations and then the information structures of the optimal filter for a continuous-time nonlinear system are made clear, which has been presented as the interesting open problems by Bucy.     

On impulse and continuous observation control design in Kalman filtering problem

This paper develops the observation control method for refining the Kalman–Bucy estimates, which is based on impulsive modeling of the transition matrix in an observation equation, thus engaging discrete-continuous observations. The impulse observation control generates on-line computable jumps of the estimate variance from its current position towards zero and, as a result, enables us to instantaneously obtain the estimate, whose variance is closer to zero. The filtering equations over impulse-controlled observations are obtained in the Kalman–Bucy filtering problem. The method for feedback design of control of the estimate variance is developed. First, the pure impulse control is used, and, next, the combination of the impulse and continuous control components is employed. The considered examples allow us to compare the properties of these control and filtering methodologies.     

Discrete-time filtering for nonlinear polynomial systems over linear observations

This paper designs a discrete-time filter for nonlinear polynomial systems driven by additive white Gaussian noises over linear observations. The solution is obtained by computing the time-update and measurement-update equations for the state estimate and the error covariance matrix. A closed form of this filter is obtained by expressing the conditional expectations of polynomial terms as functions of the estimate and the error covariance. As a particular case, a third-degree polynomial is considered to obtain the finite-dimensional filtering equations. Numerical simulations are performed for a third-degree polynomial system and an induction motor model. Performance of the designed filter is compared with the extended Kalman one to verify its effectiveness.     

带有色观测噪声的广义系统Kalman滤波器

对于带自回归滑动平均(ARMA)有色观测噪声的多传感器为广义离散随机线性系统,应用奇异值分解,将其变换为等价的两个降阶多传感器子系统,提出了广义系统多传感器信息融合状态滤波问题。为了提高精度,采用Kalman滤波方法,在线性最小方差按块对角阵最优加权融合准则下,给出了按矩阵加权解耦的分布式Kalman滤波器,可减少计算负担和改善局部滤波精度。为了计算最优加权,提出了局部滤波误差协方差阵的计算公式。一个Monte Carlo仿真例子说明了方法的有效性。     

Optimal state estimation for stochastic systems: an informationtheoretic approach

In this paper, we examine the problem of optimal state estimation or filtering in stochastic systems using an approach based on information theoretic measures. In this setting, the traditional minimum mean-square measure is compared with information theoretic measures, Kalman filtering theory is reexamined, and some new interpretations are offered. We show that for a linear Gaussian system, the Kalman filter is the optimal filter not only for the mean-square error measure, but for several information theoretic measures which are introduced in this work. For nonlinear systems, these same measures generally are in conflict with each other, and the feedback control policy has a dual role with regard to regulation and estimation. For linear stochastic systems with general noise processes, a lower bound on the achievable mutual information between the estimation error and the observation are derived. The properties of an optimal (probing) control law and the associated optimal filter, which achieve this lower bound, and their relationships are investigated. It is shown that for a linear stochastic system with an affine linear filter for the homogeneous system, under some reachability and observability conditions, zero mutual information between estimation error and observations can be achieved only when the system is Gaussian     

Relation between filter using covariance information and Kalman filter

This paper presents sequential algorithms for the optimal impulse function, Kalman gain and the error variance in linear least squares filtering problems, when the autocovariance function of the signal is given in the form of a semi-degenerate kernel, and the additive observation noise in white Gaussian. A digital simulation result indicates that the algorithms presented in this paper are feasible, and that the values of Kalman gain and the error variance calculated by these algorithms approach to those obtained by the Kalman filter theory, for time sufficiently large.     

State-estimation in linear distributed parameter systems via boundary measurements containing coloured noise

The problem of estimating the state of linear distributed parameter systems from boundary measurements is considered. The measurement noise is assumed to be coloured. The optimum filter equations are derived by the applications of innovation theory. Tt represents the boundary version of the distributed Kalman—Bucy filter with coloured measurement noise.     

具有测量数据丢失的离散不确定时滞系统鲁棒Kalman滤波

研究了具有测量数据丢失的离散不确定时滞系统鲁棒Kalman滤波问题, 其中时延存在于系统状态和观测值中. 模型的不确定性通过在系统矩阵中引入随机参数扰动来表示, 测量数据丢失现象则通过一个满足Bernoulli分布且统计特性已知的随机变量来描述. 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一个新的滤波器设计方法, 并且保证了滤波器的维数与原系统相等. 与传统的状态增广方法相比, 当时延比较大时, 该方法可以有效降低计算量. 最后, 给出一个仿真例子说明所提方法的有效性.     

带未知有色观测噪声的自校正融合Kalman滤波器

对于带未知有色观测噪声的多传感器线性离散定常随机系统, 未知模型参数和噪声方差的一致的融合估值器用递推增广最小二乘法(RELS)和求解相关函数方程得到. 将这些估值器代入到最优解耦融合Kalman滤波器中, 得出了自校正解耦融合Kalman滤波器, 并用动态方差误差系统分析(DVESA)和动态误差分析(DESA)方法证明了它收敛于最优解耦融合Kalman滤波器, 因而具有渐近最优性. 一个带3传感器跟踪系统的仿真例子说明了其有效 性.     

On convergence of the unscented Kalman–Bucy filter using contraction theory

Contraction theory entails a theoretical framework in which convergence of a nonlinear system can be analysed differentially in an appropriate contraction metric. This paper is concerned with utilising stochastic contraction theory to conclude on exponential convergence of the unscented Kalman–Bucy filter. The underlying process and measurement models of interest are Itô-type stochastic differential equations. In particular, statistical linearisation techniques are employed in a virtual–actual systems framework to establish deterministic contraction of the estimated expected mean of process values. Under mild conditions of bounded process noise, we extend the results on deterministic contraction to stochastic contraction of the estimated expected mean of the process state. It follows that for the regions of contraction, a result on convergence, and thereby incremental stability, is concluded for the unscented Kalman–Bucy filter. The theoretical concepts are illustrated in two case studies.