Pathwise approximation and simulation for the Zakai filtering equation through operator splitting

Pathwise approximate solutions to the Zakai nonlinear filtering equation are analyzed and computed through operator splitting techniques. Several examples are presented to test the performances of the operator splitting algorithms. The work of R. Glowinski was supported by the U.S. Army Research Office under Contract DAAL03-86-K-0138 and by NSF via Grant INT-8612680.     

Finite-dimensional solutions of a modified Zakai equation

In risk-sensitive control a modified Zakai equation arises which includes an extra term related to the exponential running cost. We show that a wide class of finite-dimensional solutions related to the Beneš filter is possible as long as nonlinearities in the drift are canceled in an appropriate way by terms in the running cost. The support of NSERC Grant A7964 is gratefully acknowledged.     

A quadrature-based method of moments for nonlinear filtering

According to the nonlinear filtering theory, optimal estimates of a general continuous-discrete nonlinear filtering problem can be obtained by solving the Fokker-Planck equation, coupled with a Bayesian update rule. This procedure does not rely on linearizations of the dynamical and/or measurement models. However, the lack of fast and efficient methods for solving the Fokker-Planck equation presents challenges in real time nonlinear filtering problems. In this paper, a direct quadrature method of moments is introduced to solve the Fokker-Planck equation efficiently and accurately. This approach involves representation of the state conditional probability density function in terms of a finite collection of Dirac delta functions. The weights and locations (abscissas) in this representation are determined by moment constraints and modified using the Bayes’ rule according to measurement updates. As demonstrated by numerical examples, this approach appears to be promising in the field of nonlinear filtering.     

A Zakai equation derivation of the extended Kalman filter

A discrete time filter is considered where both the observation and signal process have non-linear dynamics with additive Gaussian noise. Using the reference probability framework a convolution type Zakai equation is obtained which updates the unnormalized conditional density. Using first order approximations this equation can be solved recursively and the extended Kalman filter can be derived.     

融合偏微分方程和中值滤波的图像去噪模型

针对基于偏微分方程(PDE)的图像去噪模型不能有效地去除脉冲噪声,并且低阶偏微分方程在去噪的同时会出现“块效应”现象的问题,提出一种融合偏微分方程和自适应中值滤波的图像去噪模型。该模型通过对图像梯度的分析,在梯度变化剧烈区域和梯度变化微小区域利用二阶模型去噪以提高去噪效率;而在梯度渐变区域利用四阶模型平滑图像以避免出现“块效应”现象。同时,利用脉冲噪声梯度值远大于边缘梯度值的特点,定位脉冲噪声所在区域,在该区域利用自适应中值滤波消除脉冲噪声。该方法能有效去除脉冲噪声,保护图像边缘并消除“块效应”现象,同时提高了去噪效率。实验表明了该模型的有效性。     

Polynomial filtering for nonlinear stochastic systems with state‐ and disturbance‐dependent noises

This article is concerned with the polynomial filtering problem for a class of nonlinear stochastic systems governed by the Itô differential equation. The system under investigation involves polynomial nonlinearities, unknown‐but‐bounded disturbances, and state‐ and disturbance‐dependent noises ((x,d)‐dependent noises for short). By expanding the polynomial nonlinear functions in Taylor series around the state estimate, a new polynomial filter design method is developed with hope to reduce the conservatism of the existing results. In virtue of stochastic analysis and inequality technique, sufficient conditions in terms of parameter‐dependent linear matrix inequalities (PDLMIs) are derived to guarantee that the estimation error system is input‐to‐state stable in probability. Moreover, the desired polynomial matrix can be obtained by solving the PDLMIs via the sum‐of‐squares approach. The effectiveness and applicability of the proposed method are illustrated by two numerical examples with one concerning the permanent magnet synchronous motor.     

A note on a nonlinear semigroup for controlled partially observed diffusions

This paper is concerned with the separated control problems for optimal stochastic controls under partial observations. The ‘pathwise’ method of nonlinear filtering is extended to the case when a function h(X_tY_t, of state X_t, and observation Y_t 3plus additive white noise is observed. Markov and continuity properties of the unnormalized conditional distribution measure are obtained and the Nisio nonlinear semigroup is found.     

Existence of optimal controls for SPDE with locally monotone coefficients

The aim of this paper is to investigate the existence of optimal controls for systems described by stochastic partial differential equations (SPDEs) with locally monotone coefficients controlled by external forces which are feedback controls. To attain our objective we adapt the argument of Lisei (2002) where the existence of optimal controls to the stochastic Navier–Stokes equation was studied. The results obtained in the present paper may be applied to demonstrate the existence of optimal controls to various types of controlled SPDEs such as: a stochastic nonlocal equation and stochastic semilinear equations which are locally monotone equations; we also apply the result to a monotone equation such as the stochastic reaction–diffusion equation and to a stochastic linear equation.     

Exact solutions of a KdV equation hierarchy with variable coefficients

In this paper, a nonisospectral and variable-coefficient KdV equation hierarchy with self-consistent sources is derived from the related linear spectral problem. Exact solutions of the KdV equation hierarchy are obtained through the inverse scattering transformation (IST). It is shown that the IST is an effective mathematical tool for solving the whole hierarchy of nonisospectral nonlinear partial differential equations with self-consistent sources.     

Numerical scheme based on similarity reductions for the regularized long wave equation

Numerical solution based on similarity reductions for partial differential equations used to get the numerical scheme for the regularized long wave (RLW) equation. The similarity reductions for RLW equation are obtained locally on subdomains defined by the classical three-point stencil. The ordinary differential equation, which deduced from the similarity reduction can be linearized, integrated analytically and then obtain the solution. This approch eliminates the difficulties associated with boundary conditions for the similarity reduction over the whole solution domain. Numerical results are obtained for test problem. The computed results using our scheme confirm the accuracy of our scheme.     

数值鲁棒的非线性滤波新方法

应用多维情形的二阶插值公式构造新型非线性滤波器。该滤波器不需非线性函数的偏导计算，便能代替常规的扩展卡尔曼滤波器，并有滤波精度高、数值计算稳定和适用范围宽等优点。仿真实例表明新滤波器具有较高的性能。     

A phase-space formulation and Gaussian approximation of the filtering equations for nonlinear quantum stochastic systems

This paper is concerned with a filtering problem for a class of nonlinear quantum stochastic systems with multichannel nondemolition measurements. The system-observation dynamics are governed by a Markovian Hudson-Parthasarathy quantum stochastic differential equation driven by quantum Wiener processes of bosonic fields in vacuum state. The Hamiltonian and system-field coupling operators, as functions of the system variables, are assumed to be represented in a Weyl quantization form. Using the Wigner-Moyal phase-space framework, we obtain a stochastic integro-differential equation for the posterior quasi-characteristic function (QCF) of the system conditioned on the measurements. This equation is a spatial Fourier domain representation of the Belavkin-Kushner-Stratonovich stochastic master equation driven by the innovation process associated with the measurements. We discuss a specific form of the posterior QCF dynamics in the case of linear system-field coupling and outline a Gaussian approximation of the posterior quantum state.     

White noise theory of robust nonlinear filtering with correlated state and observation noises

In the existing ‘direct’ white noise theory of nonlinear filtering, the state process is still modelled as a Markov process satisfying an Itô stochastic differential equation, while a ‘finitely additive’ white noise is used to model the observation noise. We remove this asymmetry by modelling the state process as the solution of a (stochastic) differential equation with a ‘finitely additive’ white noise as the input. This enables us to introduce correlation between the state and observation noises, and to obtain robust nonlinear filtering equations in the correlated noise case.     

Stabilization of a class of nonlinear uncertain ordinary differential equation by parabolic partial differential equation controller

This article is concerned with stabilization for a class of uncertain nonlinear ordinary differential equation (ODE) with dynamic controller governed by linear 1?d heat partial differential equation (PDE). The control input acts at the one boundary of the heat's controller domain and the second boundary injects a Dirichlet term in ODE plant. The main contribution of this article is the use of the recent infinite‐dimensional backstepping design for state feedback stabilization design of coupled PDE‐ODE systems, to stabilize exponentially the nonlinear uncertain systems, under the restrictions that (a) the right‐hand side of the ODE equation has the classical particular form: linear controllable part with an additive nonlinear uncertain function satisfying lower triangular linear growth condition, and (b) the length of the PDE domain has to be restricted. We solve the stabilization problem despite the fact that all known backstepping transformation in the literature cannot decouple the PDE and the ODE subsystems. Such difficulty is due to the presence of a nonlinear uncertain term in the ODE system. This is done by introducing a new globally exponentially stable target system for which the PDE and ODE subsystems are strongly coupled. Finally, an example is given to illustrate the design procedure of the proposed method.     

A lower bound on the integral of the nonlinear filtering error

A lower bound on the nonlinear filtering error derived on the basis of information theoretic considerations will be proposed. The tightness of this bound will be shown in an example.     

有色量测噪声下带广义未知扰动的递推上限滤波

针对博弈对抗环境下利用快速采样雷达进行非合作目标跟踪带来的有色噪声和未知干扰共存问题, 本文提出有色量测噪声下带广义未知扰动的随机动态系统递推上限滤波. 这里, 有色量测噪声用于描述由于快速采样或持续干扰带来的噪声相关性, 广义未知扰动用于建模博弈对抗对雷达观测带来的异常影响(先验信息缺失). 针对所考虑系统, 通过参数优化实现状态估计误差协方差上限(而不是理论值)的在线递推, 提出有色噪声下上限滤波(CU-BF), 给出状态估计误差协方差最小上限的近似实现, 讨论了所提CUBF的存在性条件. 在具有时变未知扰动和有色量测噪声的目标跟踪仿真中验证了所提方法的有效性.