Approximate finite-dimensional filtering for polynomial states over polynomial observations

In this article, the mean-square filtering problem for polynomial system states over polynomial observations is studied proceeding from the general expression for the stochastic Ito differentials of the mean-square estimate and the error variance. In contrast to the previously obtained results, this article deals with the general case of nonlinear polynomial states and observations. As a result, the Ito differentials for the mean-square estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining an approximate closed-form finite-dimensional system of the filtering equations for any polynomial state over observations with any polynomial drift is then established. In the example, the obtained closed-form filter is applied to solve the third-order sensor filtering problem for a quadratic state, assuming a conditionally Gaussian initial condition for the extended third-order state vector. The simulation results show that the designed filter yields a reliable and rapidly converging estimate.     

Root-mean-square filtering of the state of polynomial stochastic systems with multiplicative noise

Some results obtained by the present author in the field of designing the finitedimensional root-mean-square filters for stochastic systems with polynomial equations of state and multiplicative noise from the linear observations were overviewed. A procedure to derive the finite-dimensional system of approximate filtering equations for a polynomial arbitrary-order equation of state was presented. The closed system of filtering equations for the root-mean-square estimate and covariance matrix error was deduced explicitly for special cases of linear and quadratic coefficients of drift and diffusion in the equation of state. For linear stochastic systems with unknown parameters, the problem of joint root-mean-square state filtering and identification of the parameters from linear observations was considered in the Appendix.     

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.     

A continuous-time reduced-order filter for estimating the state vector of a linear stochastic system†

A design procedure for a, reduced-order filter is suggested and the differential equation obeyed by the estimate error covariance matrix is obtained. The problem of optimizing the filter within the design framework is partially solved.     

Quantised polynomial filtering for nonlinear systems with missing measurements

This paper is concerned with the polynomial filtering problem for a class of nonlinear systems with quantisations and missing measurements. The nonlinear functions are approximated with polynomials of a chosen degree and the approximation errors are described as low-order polynomial terms with norm-bounded coefficients. The transmitted outputs are quantised by a logarithmic quantiser and are also subject to randomly missing measurements governed by a Bernoulli distributed sequence taking values on 0 or 1. Dedicated efforts are made to derive an upper bound of the filtering error covariance in the simultaneous presence of the polynomial approximation errors, the quantisations as well as the missing measurements at each time instant. Such an upper bound is then minimised through designing a suitable filter gain by solving a set of matrix equations. The filter design algorithm is recursive and therefore applicable for online computation. An illustrative example is exploited to show the effectiveness of the proposed algorithm.     

基于相关向量机的小世界神经元网络拓扑估计

采用相关向量机从含噪时间序列中估计小世界神经元网络的节点动力学方程和拓扑结构。在具有多项式结构或能以幂级数展开的动力学系统中,将未知方程写成统一多项式形式,原动力学方程的项在统一多项式中是稀疏的,利用稀疏贝叶斯学习估计出稀疏项从而实现动力学方程和拓扑结构的估计。利用该方法对FHN小世界神经元网络进行节点动力学方程和拓扑估计,结果表明,该方法能快速准确地估计节点动力学方程结构和网络拓扑,对动力学方程系数和网络耦合强度有很高的估计精度,而且对噪声有强鲁棒性。     

基于单领航者相对位置测量的多AUV协同导航系统定位性能分析

自主水下航行器 (Autonomous underwater vehicle, AUV) 的协同导航是解决水下导航定位问题的重要方法, 其中导航系统的定位误差增长特性是衡量其定位性能的关键指标. 本文针对单领航者相对位置测量的多 AUV 协同导航系统, 利用扩展卡尔曼滤波方法建立了导航系统的整体定位误差关于相对位置量测误差的传递方程. 在此基础上, 通过求解系统定位误差随时间演化的代数黎卡提方程, 得到了其在稳态情形下的方差上界估计. 理论分析表明, 单领航 AUV 协同导航系统的整体定位误差有界收敛且与初始化滤波方差无关, 具有良好的综合性能. 最后, 仿真实例验证了文中理论分析结果的正确性.     

Stochastic stability of cubature predictive filterCubature 预测滤波的随机稳定性分析

In this paper, the cubature predictive filter (CPF) is derived based on a third-degree spherical-radial cubature rule. It provides a set of cubature-points scaling linearly with the state-vector dimension, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear predictive filter (PF). In order to facilitate the new method, the algorithm CPF is given firstly. Then, the theoretical analyses demonstrate that the estimated accuracy of the model error and system for the proposed CPF is higher than that of the traditional PF. Moreover, the authors analyze the stochastic boundedness and the error behavior of CPF for general nonlinear systems in a stochastic framework. In particular, the theoretical results present that the estimation error remains bounded and the covariance keeps stable if the system’s initial estimation error, disturbing noise terms as well as the model error are small enough, which is the core part of the CPF theory. All of the results have been demonstrated by numerical simulations for a nonlinear example system.     

随机逼近自适应滤波在捷联惯导系统初始对准中的应用

对于测量噪声方差未知的捷联惯导系统(SINS),采用常规Kalman滤波进行初始对准会造成较大状态估计误差,甚至使滤波器发散。为了解决系统测量噪声方差未知或不确切知道时SINS的误差估计问题,提出一种基于随机逼近的自适应滤波方法。该方法将Robbins-Monro算法与Kalman滤波相结合,通过简化求逆运算,解决了系统观测噪声特性未知情况下SINS的误差估计问题,并提高了算法的数值稳定性。仿真结果表明,该方法能在系统测量噪声方差未知情况下有效实现SINS初始对准。     

Optimal filtering for linear systems with state and observation delays

In this paper, the optimal filtering problem for linear systems with state and observation delays is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate, error variance, and various error covariances. 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 filter gain matrix consists, in the general case, of an infinite set of equations. It is however demonstrated that a finite set of the filtering equations, whose number is specified by the ratio between the current filtering horizon and the delay values, can be obtained in the particular case of equal or commensurable (τ=qh, q is natural) delays in the observation and state equations. In the example, performance of the designed optimal filter for linear systems with state and observation delays 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. Copyright © 2005 John Wiley & Sons, Ltd.     

Optimality of an error estimate of a one-step numerical integration method for certain initial value problems

In a recent paper, an error estimate of a one-step numerical method, originated from the Lanczos tau method, for initial value problems for first order linear ordinary differential equations with polynomial coefficients, was obtained, based on the error of the Lanczos econo-mization process. Numerical results then revealed that the estimate gives, correctly, the order of the tau approximant being sought. In the present paper we further establish that the error estimate is optimum with respect to the integration of the error equation. Numerical examples are included for completeness.     

量测随机延迟下带相关乘性噪声的非线性系统分布式估计

本文提出了乘性噪声和加性噪声相关下的量测随机延迟非线性系统分布式状态估计.在所考虑系统中,相关状态被多传感器簇构成的传感器网所观测.所得理想量测被传送到远程分布式处理网,并伴随服从一阶马尔可夫过程的随机延迟.在此基础上,本文提出了分布式高斯信息滤波(distributed Gaussian-information filter,DGIF),来实现估计精度与计算时间的折中.在单处理节点/单元中,以估计误差协方差最小化为准则,设计了相应的高斯递推滤波,并实现了延迟概率的在线递推估计.进一步地,在分布式处理网中,基于非线性量测方程的统计线性回归,结合一致性算法,给出了一种分布式信息滤波形式,有效实现了分布式融合.分别在单处理单元和分布式处理网中仿真验证了所提算法的有效性.     

Estimation in linear discrete systems with multiple time delays

The method of orthogonal projection is used to derive the equations for optimally estimating the state of a nonstationary linear discrete system with multiple time delays. A Kalman-type filter is developed, along with the necessary recursive error and cross error covariance matrix equations. A numerical example is included.     

Quantized innovations Kalman filter: stability and modification with scaling quantization

The stability of quantized innovations Kalman filtering (QIKF) is analyzed. In the analysis, the correlation between quantization errors and measurement noises is considered. By taking the quantization errors as a random perturbation in the observation system, the QIKF for the original system is equivalent to a Kalman-like filtering for the equivalent state-observation system. Thus, the estimate error covariance matrix of QIKF can be more exactly analyzed. The boundedness of the estimate error covariance matrix of QIKF is obtained under some weak conditions. The design of the number of quantized levels is discussed to guarantee the stability of QIKF. To overcome the instability and divergence of QIKF when the number of quantization levels is small, we propose a Kalman filter using scaling quantized innovations. Numerical simulations show the validity of the theorems and algorithms.     

Smoothing for multipoint boundary value models

We show how to solve the linear least-squares smoothing problem for multipoint boundary value models. The complementary model is derived and is used to determine the Hamiltonian equations for the smoothed state estimate and its error covariance. Stable algorithms are obtained using an invariant imbedding/multiple shooting procedure.     

Error analysis of the Chebyshev collocation method for linear second-order partial differential equations

The purpose of this study is to apply the Chebyshev collocation method to linear second-order partial differential equations (PDEs) under the most general conditions. The method is given with a priori error estimate which is obtained by polynomial interpolation. The residual correction procedure is modified to the problem so that the absolute error may be estimated. Finally, the effectiveness of the method is illustrated in several numerical experiments such as Laplace and Poisson equations. Numerical results are overlapped with the theoretical results.     

A heuristic Kalman filter for a class of nonlinear systems

One of the basic assumptions involved in the "optimality" of the Kalman filter theory is that the system under consideration must be linear. If the model is nonlinear, a linearization procedure is usually performed in deriving the filtering equations. This approach requires the nonlinear system dynamics to be differentiable. This note is an attempt to develop a heuristic Kalman filter for a class of nonlinear systems, with bounded first-order growth, that does not require the system dynamics to be differentiable. The proposed filter approximates the nonlinear state function by its state argument multiplied by a particular gain matrix only in the recursion of the estimation error covariance matrix. Under certain conditions, the error covariance remains bounded by bounds which can be precomputed from noise and system models, and the upper bound tends to zero when the state noise covariance tends to zero. A numerical example, with backlash nonlinearity, is also added.     

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.     

Specific optimal estimation

The goal of the specific optimal estimation problem is to achieve near-optimal (minimum variance) estimation using a structure that is easier to implement than the optimal solution. One chooses a reasonable configuration for the filter in which certain parameters are unspecified and then selects the parameters so that its performance is optimized. The problem is formulated as a two-point boundary-value problem resulting from consideration of the covariance of error of the estimate and application of the matrix-minimum principle. The examples presented indicate that near-optimal results can be obtained using a filter designed in this way.     

Mean-square filter design for stochastic polynomial systems with Gaussian and Poisson noises

This paper addresses the mean-square finite-dimensional filtering problem for polynomial system states with both, Gaussian and Poisson, white noises over linear observations. A constructive procedure is established to design the mean-square filtering equations for system states described by polynomial equations of an arbitrary finite degree. An explicit closed form of the designed filter is obtained in case of a third-order polynomial system. The theoretical result is complemented with an illustrative example verifying performance of the designed filter.